sets - university of british columbia department of ...elyse/220/2016/1sets.pdf · 1. sets 1.1...
TRANSCRIPT
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Sets
First chapter: Sets
Slow start–introduction to sets, easy problems
Lead to more in-depth problems.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Sets
First chapter: Sets
Slow start–introduction to sets, easy problems
Lead to more in-depth problems.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Sets
First chapter: Sets
Slow start–introduction to sets, easy problems
Lead to more in-depth problems.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definitions
Definition
A set is a collection of things. The things in the set are called elements ofthe set.
Convention
We often write the elements of a set separated by commas inside curly braces.
Example:
{polar, brown, N.Am. black, Asiatic black, Andean, panda, sun, sloth}
Set: extant species of bear.
This is a finite set.It has a finite number of elements.
Example:P = {2, 3, 5, 7, 11, 13, 17, 19, . . .}
Ellipsis (. . .) means “continue the pattern.”
This is an infinite set. It has an infinite number of elements.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definitions
Definition
A set is a collection of things. The things in the set are called elements ofthe set.
Convention
We often write the elements of a set separated by commas inside curly braces.
Example:
{polar, brown, N.Am. black, Asiatic black, Andean, panda, sun, sloth}
Set: extant species of bear.
This is a finite set.It has a finite number of elements.
Example:P = {2, 3, 5, 7, 11, 13, 17, 19, . . .}
Ellipsis (. . .) means “continue the pattern.”
This is an infinite set. It has an infinite number of elements.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definitions
Definition
A set is a collection of things. The things in the set are called elements ofthe set.
Convention
We often write the elements of a set separated by commas inside curly braces.
Example:
{polar, brown, N.Am. black, Asiatic black, Andean, panda, sun, sloth}
Set: extant species of bear.
This is a finite set.It has a finite number of elements.
Example:P = {2, 3, 5, 7, 11, 13, 17, 19, . . .}
Ellipsis (. . .) means “continue the pattern.”
This is an infinite set. It has an infinite number of elements.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definitions
Definition
A set is a collection of things. The things in the set are called elements ofthe set.
Convention
We often write the elements of a set separated by commas inside curly braces.
Example:
{polar, brown, N.Am. black, Asiatic black, Andean, panda, sun, sloth}
Set: extant species of bear.
This is a finite set.It has a finite number of elements.
Example:P = {2, 3, 5, 7, 11, 13, 17, 19, . . .}
Ellipsis (. . .) means “continue the pattern.”
This is an infinite set. It has an infinite number of elements.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definitions
Definition
A set is a collection of things. The things in the set are called elements ofthe set.
Convention
We often write the elements of a set separated by commas inside curly braces.
Example:
{polar, brown, N.Am. black, Asiatic black, Andean, panda, sun, sloth}
Set: extant species of bear. This is a finite set.It has a finite number of elements.
Example:P = {2, 3, 5, 7, 11, 13, 17, 19, . . .}
Ellipsis (. . .) means “continue the pattern.”
This is an infinite set. It has an infinite number of elements.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definitions
Definition
A set is a collection of things. The things in the set are called elements ofthe set.
Convention
We often write the elements of a set separated by commas inside curly braces.
Example:
{polar, brown, N.Am. black, Asiatic black, Andean, panda, sun, sloth}
Set: extant species of bear. This is a finite set.It has a finite number of elements.
Example:P = {2, 3, 5, 7, 11, 13, 17, 19, . . .}
Ellipsis (. . .) means “continue the pattern.”This is an infinite set. It has an infinite number of elements.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definitions
Definition
Two sets are equal if they contain exactly the same elements.
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}
“element of” symbol
We write 10 ∈ A to mean “10 is an element of A”.
We write 10 6∈ A to mean “10 is not an element of A”.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definitions
Definition
Two sets are equal if they contain exactly the same elements.
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}
“element of” symbol
We write 10 ∈ A to mean “10 is an element of A”.
We write 10 6∈ A to mean “10 is not an element of A”.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definitions
Definition
Two sets are equal if they contain exactly the same elements.
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}
“element of” symbol
We write 10 ∈ A to mean “10 is an element of A”.
We write 10 6∈ A to mean “10 is not an element of A”.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definitions
Definition
Two sets are equal if they contain exactly the same elements.
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}
“element of” symbol
We write 10 ∈ A to mean “10 is an element of A”.
We write 10 6∈ A to mean “10 is not an element of A”.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definitions
Definition
Two sets are equal if they contain exactly the same elements.
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}
“element of” symbol
We write 10 ∈ A to mean “10 is an element of A”.
We write 10 6∈ A to mean “10 is not an element of A”.
How can you finish these?
0 ∈
B
7
6∈
A
35, 45, 55 ∈
E This is a common “abuse of notation”
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definitions
Definition
Two sets are equal if they contain exactly the same elements.
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}
“element of” symbol
We write 10 ∈ A to mean “10 is an element of A”.
We write 10 6∈ A to mean “10 is not an element of A”.
How can you finish these?
0 ∈ B
7
6∈
A
35, 45, 55 ∈
E This is a common “abuse of notation”
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definitions
Definition
Two sets are equal if they contain exactly the same elements.
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}
“element of” symbol
We write 10 ∈ A to mean “10 is an element of A”.
We write 10 6∈ A to mean “10 is not an element of A”.
How can you finish these?
0 ∈ B
7 6∈ A
35, 45, 55 ∈
E This is a common “abuse of notation”
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definitions
Definition
Two sets are equal if they contain exactly the same elements.
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}
“element of” symbol
We write 10 ∈ A to mean “10 is an element of A”.
We write 10 6∈ A to mean “10 is not an element of A”.
How can you finish these?
0 ∈ B
7 6∈ A
35, 45, 55 ∈ E
This is a common “abuse of notation”
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definitions
Definition
Two sets are equal if they contain exactly the same elements.
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}
“element of” symbol
We write 10 ∈ A to mean “10 is an element of A”.
We write 10 6∈ A to mean “10 is not an element of A”.
How can you finish these?
0 ∈ B
7 6∈ A
35, 45, 55 ∈ E This is a common “abuse of notation”
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definitions
Definition
Two sets are equal if they contain exactly the same elements.
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}
“element of” symbol
We write 10 ∈ A to mean “10 is an element of A”.
We write 10 6∈ A to mean “10 is not an element of A”.
We write |A| to mean the size, or cardinality of a set1 A. For finite sets,
this means the number of distinct elements.
1For now, we’ll only talk about the cardinalities of finite sets
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definitions
Definition
Two sets are equal if they contain exactly the same elements.
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}
“element of” symbol
We write 10 ∈ A to mean “10 is an element of A”.
We write 10 6∈ A to mean “10 is not an element of A”.
We write |A| to mean the size, or cardinality of a set1 A. For finite sets,
this means the number of distinct elements.
How can you finish these?
|A| =
4
|A| =∣∣∣
D
∣∣∣1For now, we’ll only talk about the cardinalities of finite sets
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definitions
Definition
Two sets are equal if they contain exactly the same elements.
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}
“element of” symbol
We write 10 ∈ A to mean “10 is an element of A”.
We write 10 6∈ A to mean “10 is not an element of A”.
We write |A| to mean the size, or cardinality of a set1 A. For finite sets,
this means the number of distinct elements.
How can you finish these?
|A| = 4
|A| =∣∣∣
D
∣∣∣1For now, we’ll only talk about the cardinalities of finite sets
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definitions
Definition
Two sets are equal if they contain exactly the same elements.
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}
“element of” symbol
We write 10 ∈ A to mean “10 is an element of A”.
We write 10 6∈ A to mean “10 is not an element of A”.
We write |A| to mean the size, or cardinality of a set1 A. For finite sets,
this means the number of distinct elements.
How can you finish these?
|A| = 4
|A| =∣∣∣ D ∣∣∣
1For now, we’ll only talk about the cardinalities of finite sets
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definitions
Definition
Two sets are equal if they contain exactly the same elements.
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}
“element of” symbol
We write 10 ∈ A to mean “10 is an element of A”.
We write 10 6∈ A to mean “10 is not an element of A”.
We write |A| to mean the size, or cardinality of a set1 A. For finite sets,
this means the number of distinct elements.
True or false: C=3
False. C is a set, 3 is a number.
True or false: If two sets are equal, their cardinalities are equal.
True.
1For now, we’ll only talk about the cardinalities of finite sets
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definitions
Definition
Two sets are equal if they contain exactly the same elements.
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}
“element of” symbol
We write 10 ∈ A to mean “10 is an element of A”.
We write 10 6∈ A to mean “10 is not an element of A”.
We write |A| to mean the size, or cardinality of a set1 A. For finite sets,
this means the number of distinct elements.
True or false: C=3 False. C is a set, 3 is a number.
True or false: If two sets are equal, their cardinalities are equal.
True.
1For now, we’ll only talk about the cardinalities of finite sets
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definitions
Definition
Two sets are equal if they contain exactly the same elements.
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}
“element of” symbol
We write 10 ∈ A to mean “10 is an element of A”.
We write 10 6∈ A to mean “10 is not an element of A”.
We write |A| to mean the size, or cardinality of a set1 A. For finite sets,
this means the number of distinct elements.
True or false: C=3 False. C is a set, 3 is a number.
True or false: If two sets are equal, their cardinalities are equal.
True.1For now, we’ll only talk about the cardinalities of finite sets
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Special Sets
the Empty Set, ∅ . This is the set with no elements.
Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| = 0True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”
the Natural Numbers, N .
N = {1, 2, 3, 4, 5, . . .}
This is an infinite set.∞ 6∈ NControversy: is 0 a natural number? (In our text, it is not.)
the Integers, Z .
Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}
No controversy: 0 is definitely an integerZ stands for “number” if you’re German (Zahl)Which is bigger, N or Z? stay tuned
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Special Sets
the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.
∅ = {}|∅| = 0True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”
the Natural Numbers, N .
N = {1, 2, 3, 4, 5, . . .}
This is an infinite set.∞ 6∈ NControversy: is 0 a natural number? (In our text, it is not.)
the Integers, Z .
Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}
No controversy: 0 is definitely an integerZ stands for “number” if you’re German (Zahl)Which is bigger, N or Z? stay tuned
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Special Sets
the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}
|∅| = 0True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”
the Natural Numbers, N .
N = {1, 2, 3, 4, 5, . . .}
This is an infinite set.∞ 6∈ NControversy: is 0 a natural number? (In our text, it is not.)
the Integers, Z .
Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}
No controversy: 0 is definitely an integerZ stands for “number” if you’re German (Zahl)Which is bigger, N or Z? stay tuned
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Special Sets
the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| =
0True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”
the Natural Numbers, N .
N = {1, 2, 3, 4, 5, . . .}
This is an infinite set.∞ 6∈ NControversy: is 0 a natural number? (In our text, it is not.)
the Integers, Z .
Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}
No controversy: 0 is definitely an integerZ stands for “number” if you’re German (Zahl)Which is bigger, N or Z? stay tuned
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Special Sets
the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| = 0
True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”
the Natural Numbers, N .
N = {1, 2, 3, 4, 5, . . .}
This is an infinite set.∞ 6∈ NControversy: is 0 a natural number? (In our text, it is not.)
the Integers, Z .
Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}
No controversy: 0 is definitely an integerZ stands for “number” if you’re German (Zahl)Which is bigger, N or Z? stay tuned
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Special Sets
the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| = 0True or false: If |A| = 0, then A = ∅.
True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”
the Natural Numbers, N .
N = {1, 2, 3, 4, 5, . . .}
This is an infinite set.∞ 6∈ NControversy: is 0 a natural number? (In our text, it is not.)
the Integers, Z .
Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}
No controversy: 0 is definitely an integerZ stands for “number” if you’re German (Zahl)Which is bigger, N or Z? stay tuned
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Special Sets
the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| = 0True or false: If |A| = 0, then A = ∅. True.
True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”
the Natural Numbers, N .
N = {1, 2, 3, 4, 5, . . .}
This is an infinite set.∞ 6∈ NControversy: is 0 a natural number? (In our text, it is not.)
the Integers, Z .
Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}
No controversy: 0 is definitely an integerZ stands for “number” if you’re German (Zahl)Which is bigger, N or Z? stay tuned
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Special Sets
the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| = 0True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.
False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”
the Natural Numbers, N .
N = {1, 2, 3, 4, 5, . . .}
This is an infinite set.∞ 6∈ NControversy: is 0 a natural number? (In our text, it is not.)
the Integers, Z .
Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}
No controversy: 0 is definitely an integerZ stands for “number” if you’re German (Zahl)Which is bigger, N or Z? stay tuned
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Special Sets
the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| = 0True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.
“An empty box is not the same as a box with an empty box inside it.”
the Natural Numbers, N .
N = {1, 2, 3, 4, 5, . . .}
This is an infinite set.∞ 6∈ NControversy: is 0 a natural number? (In our text, it is not.)
the Integers, Z .
Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}
No controversy: 0 is definitely an integerZ stands for “number” if you’re German (Zahl)Which is bigger, N or Z? stay tuned
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Special Sets
the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| = 0True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”
∅ {∅}
the Natural Numbers, N .
N = {1, 2, 3, 4, 5, . . .}
This is an infinite set.∞ 6∈ NControversy: is 0 a natural number? (In our text, it is not.)
the Integers, Z .
Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}
No controversy: 0 is definitely an integerZ stands for “number” if you’re German (Zahl)Which is bigger, N or Z? stay tuned
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Special Sets
the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| = 0True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”
the Natural Numbers, N .
N = {1, 2, 3, 4, 5, . . .}
This is an infinite set.∞ 6∈ NControversy: is 0 a natural number? (In our text, it is not.)
the Integers, Z .
Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}
No controversy: 0 is definitely an integerZ stands for “number” if you’re German (Zahl)Which is bigger, N or Z? stay tuned
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Special Sets
the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| = 0True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”
the Natural Numbers, N .
N = {1, 2, 3, 4, 5, . . .}
This is an infinite set.
∞ 6∈ NControversy: is 0 a natural number? (In our text, it is not.)
the Integers, Z .
Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}
No controversy: 0 is definitely an integerZ stands for “number” if you’re German (Zahl)Which is bigger, N or Z? stay tuned
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Special Sets
the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| = 0True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”
the Natural Numbers, N .
N = {1, 2, 3, 4, 5, . . .}
This is an infinite set.∞ 6∈ N
Controversy: is 0 a natural number? (In our text, it is not.)
the Integers, Z .
Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}
No controversy: 0 is definitely an integerZ stands for “number” if you’re German (Zahl)Which is bigger, N or Z? stay tuned
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Special Sets
the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| = 0True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”
the Natural Numbers, N .
N = {1, 2, 3, 4, 5, . . .}
This is an infinite set.∞ 6∈ NControversy: is 0 a natural number? (In our text, it is not.)
the Integers, Z .
Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}
No controversy: 0 is definitely an integerZ stands for “number” if you’re German (Zahl)Which is bigger, N or Z? stay tuned
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Special Sets
the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| = 0True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”
the Natural Numbers, N .
N = {1, 2, 3, 4, 5, . . .}
This is an infinite set.∞ 6∈ NControversy: is 0 a natural number? (In our text, it is not.)
the Integers, Z .
Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}
No controversy: 0 is definitely an integerZ stands for “number” if you’re German (Zahl)Which is bigger, N or Z? stay tuned
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Special Sets
the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| = 0True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”
the Natural Numbers, N .
N = {1, 2, 3, 4, 5, . . .}
This is an infinite set.∞ 6∈ NControversy: is 0 a natural number? (In our text, it is not.)
the Integers, Z .
Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}
No controversy: 0 is definitely an integer
Z stands for “number” if you’re German (Zahl)Which is bigger, N or Z? stay tuned
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Special Sets
the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| = 0True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”
the Natural Numbers, N .
N = {1, 2, 3, 4, 5, . . .}
This is an infinite set.∞ 6∈ NControversy: is 0 a natural number? (In our text, it is not.)
the Integers, Z .
Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}
No controversy: 0 is definitely an integerZ stands for “number” if you’re German (Zahl)
Which is bigger, N or Z? stay tuned
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Special Sets
the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| = 0True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”
the Natural Numbers, N .
N = {1, 2, 3, 4, 5, . . .}
This is an infinite set.∞ 6∈ NControversy: is 0 a natural number? (In our text, it is not.)
the Integers, Z .
Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}
No controversy: 0 is definitely an integerZ stands for “number” if you’re German (Zahl)Which is bigger, N or Z?
stay tuned
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Special Sets
the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| = 0True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”
the Natural Numbers, N .
N = {1, 2, 3, 4, 5, . . .}
This is an infinite set.∞ 6∈ NControversy: is 0 a natural number? (In our text, it is not.)
the Integers, Z .
Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}
No controversy: 0 is definitely an integerZ stands for “number” if you’re German (Zahl)Which is bigger, N or Z? stay tuned
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Set-Builder Notation
A︸︷︷︸name
= { expression︸ ︷︷ ︸all things of the form
: rule︸︷︷︸such that
}
Examples:
{n : n is prime} (set of all primes)
E = {2n : n ∈ N}= {2(1), 2(2), 2(3), 2(4), . . .}= {2, 4, 6, 8, . . .}Positive even numbers
{n ∈ N :√n ∈ N} set of all perfect squares
Set of all odd numbers:
{2n + 1 : n ∈ Z}
All integers not divisible by 3:
{n ∈ Z :
n
36∈ Z
}
True or false: the following sets are equal.
{ −2n : n ∈ Z and −2n > 7 }{ 2(n + 1) : n ∈ N and n > 2 }{ 2n : n ∈ Z and n ≥ 4 }
True: {8, 10, 12, 14, 16, 18, . . .}
{a− b : a, b ∈ N} =
Z
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Set-Builder Notation
A︸︷︷︸name
= { expression︸ ︷︷ ︸all things of the form
: rule︸︷︷︸such that
}
Examples:
{n : n is prime}
(set of all primes)
E = {2n : n ∈ N}= {2(1), 2(2), 2(3), 2(4), . . .}= {2, 4, 6, 8, . . .}Positive even numbers
{n ∈ N :√n ∈ N} set of all perfect squares
Set of all odd numbers:
{2n + 1 : n ∈ Z}
All integers not divisible by 3:
{n ∈ Z :
n
36∈ Z
}
True or false: the following sets are equal.
{ −2n : n ∈ Z and −2n > 7 }{ 2(n + 1) : n ∈ N and n > 2 }{ 2n : n ∈ Z and n ≥ 4 }
True: {8, 10, 12, 14, 16, 18, . . .}
{a− b : a, b ∈ N} =
Z
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Set-Builder Notation
A︸︷︷︸name
= { expression︸ ︷︷ ︸all things of the form
: rule︸︷︷︸such that
}
Examples:
{n : n is prime} (set of all primes)
E = {2n : n ∈ N}
= {2(1), 2(2), 2(3), 2(4), . . .}= {2, 4, 6, 8, . . .}Positive even numbers
{n ∈ N :√n ∈ N} set of all perfect squares
Set of all odd numbers:
{2n + 1 : n ∈ Z}
All integers not divisible by 3:
{n ∈ Z :
n
36∈ Z
}
True or false: the following sets are equal.
{ −2n : n ∈ Z and −2n > 7 }{ 2(n + 1) : n ∈ N and n > 2 }{ 2n : n ∈ Z and n ≥ 4 }
True: {8, 10, 12, 14, 16, 18, . . .}
{a− b : a, b ∈ N} =
Z
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Set-Builder Notation
A︸︷︷︸name
= { expression︸ ︷︷ ︸all things of the form
: rule︸︷︷︸such that
}
Examples:
{n : n is prime} (set of all primes)
E = {2n : n ∈ N}= {2(1), 2(2), 2(3), 2(4), . . .}
= {2, 4, 6, 8, . . .}Positive even numbers
{n ∈ N :√n ∈ N} set of all perfect squares
Set of all odd numbers:
{2n + 1 : n ∈ Z}
All integers not divisible by 3:
{n ∈ Z :
n
36∈ Z
}
True or false: the following sets are equal.
{ −2n : n ∈ Z and −2n > 7 }{ 2(n + 1) : n ∈ N and n > 2 }{ 2n : n ∈ Z and n ≥ 4 }
True: {8, 10, 12, 14, 16, 18, . . .}
{a− b : a, b ∈ N} =
Z
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Set-Builder Notation
A︸︷︷︸name
= { expression︸ ︷︷ ︸all things of the form
: rule︸︷︷︸such that
}
Examples:
{n : n is prime} (set of all primes)
E = {2n : n ∈ N}= {2(1), 2(2), 2(3), 2(4), . . .}= {2, 4, 6, 8, . . .}
Positive even numbers
{n ∈ N :√n ∈ N} set of all perfect squares
Set of all odd numbers:
{2n + 1 : n ∈ Z}
All integers not divisible by 3:
{n ∈ Z :
n
36∈ Z
}
True or false: the following sets are equal.
{ −2n : n ∈ Z and −2n > 7 }{ 2(n + 1) : n ∈ N and n > 2 }{ 2n : n ∈ Z and n ≥ 4 }
True: {8, 10, 12, 14, 16, 18, . . .}
{a− b : a, b ∈ N} =
Z
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Set-Builder Notation
A︸︷︷︸name
= { expression︸ ︷︷ ︸all things of the form
: rule︸︷︷︸such that
}
Examples:
{n : n is prime} (set of all primes)
E = {2n : n ∈ N}= {2(1), 2(2), 2(3), 2(4), . . .}= {2, 4, 6, 8, . . .}Positive even numbers
{n ∈ N :√n ∈ N} set of all perfect squares
Set of all odd numbers:
{2n + 1 : n ∈ Z}
All integers not divisible by 3:
{n ∈ Z :
n
36∈ Z
}
True or false: the following sets are equal.
{ −2n : n ∈ Z and −2n > 7 }{ 2(n + 1) : n ∈ N and n > 2 }{ 2n : n ∈ Z and n ≥ 4 }
True: {8, 10, 12, 14, 16, 18, . . .}
{a− b : a, b ∈ N} =
Z
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Set-Builder Notation
A︸︷︷︸name
= { expression︸ ︷︷ ︸all things of the form
: rule︸︷︷︸such that
}
Examples:
{n : n is prime} (set of all primes)
E = {2n : n ∈ N}= {2(1), 2(2), 2(3), 2(4), . . .}= {2, 4, 6, 8, . . .}Positive even numbers
{n ∈ N :√n ∈ N}
set of all perfect squares
Set of all odd numbers:
{2n + 1 : n ∈ Z}
All integers not divisible by 3:
{n ∈ Z :
n
36∈ Z
}
True or false: the following sets are equal.
{ −2n : n ∈ Z and −2n > 7 }{ 2(n + 1) : n ∈ N and n > 2 }{ 2n : n ∈ Z and n ≥ 4 }
True: {8, 10, 12, 14, 16, 18, . . .}
{a− b : a, b ∈ N} =
Z
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Set-Builder Notation
A︸︷︷︸name
= { expression︸ ︷︷ ︸all things of the form
: rule︸︷︷︸such that
}
Examples:
{n : n is prime} (set of all primes)
E = {2n : n ∈ N}= {2(1), 2(2), 2(3), 2(4), . . .}= {2, 4, 6, 8, . . .}Positive even numbers
{n ∈ N :√n ∈ N} set of all perfect squares
Set of all odd numbers:
{2n + 1 : n ∈ Z}
All integers not divisible by 3:
{n ∈ Z :
n
36∈ Z
}
True or false: the following sets are equal.
{ −2n : n ∈ Z and −2n > 7 }{ 2(n + 1) : n ∈ N and n > 2 }{ 2n : n ∈ Z and n ≥ 4 }
True: {8, 10, 12, 14, 16, 18, . . .}
{a− b : a, b ∈ N} =
Z
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Set-Builder Notation
A︸︷︷︸name
= { expression︸ ︷︷ ︸all things of the form
: rule︸︷︷︸such that
}
Examples:
{n : n is prime} (set of all primes)
E = {2n : n ∈ N}= {2(1), 2(2), 2(3), 2(4), . . .}= {2, 4, 6, 8, . . .}Positive even numbers
{n ∈ N :√n ∈ N} set of all perfect squares
Set of all odd numbers:
{2n + 1 : n ∈ Z}
All integers not divisible by 3:
{n ∈ Z :
n
36∈ Z
}
True or false: the following sets are equal.
{ −2n : n ∈ Z and −2n > 7 }{ 2(n + 1) : n ∈ N and n > 2 }{ 2n : n ∈ Z and n ≥ 4 }
True: {8, 10, 12, 14, 16, 18, . . .}
{a− b : a, b ∈ N} =
Z
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Set-Builder Notation
A︸︷︷︸name
= { expression︸ ︷︷ ︸all things of the form
: rule︸︷︷︸such that
}
Examples:
{n : n is prime} (set of all primes)
E = {2n : n ∈ N}= {2(1), 2(2), 2(3), 2(4), . . .}= {2, 4, 6, 8, . . .}Positive even numbers
{n ∈ N :√n ∈ N} set of all perfect squares
Set of all odd numbers:
{2n + 1 : n ∈ Z}
All integers not divisible by 3:
{n ∈ Z :
n
36∈ Z
}True or false: the following sets are equal.
{ −2n : n ∈ Z and −2n > 7 }{ 2(n + 1) : n ∈ N and n > 2 }{ 2n : n ∈ Z and n ≥ 4 }
True: {8, 10, 12, 14, 16, 18, . . .}
{a− b : a, b ∈ N} =
Z
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Set-Builder Notation
A︸︷︷︸name
= { expression︸ ︷︷ ︸all things of the form
: rule︸︷︷︸such that
}
Examples:
{n : n is prime} (set of all primes)
E = {2n : n ∈ N}= {2(1), 2(2), 2(3), 2(4), . . .}= {2, 4, 6, 8, . . .}Positive even numbers
{n ∈ N :√n ∈ N} set of all perfect squares
Set of all odd numbers:
{2n + 1 : n ∈ Z}
All integers not divisible by 3:
{n ∈ Z :
n
36∈ Z
}
True or false: the following sets are equal.
{ −2n : n ∈ Z and −2n > 7 }{ 2(n + 1) : n ∈ N and n > 2 }{ 2n : n ∈ Z and n ≥ 4 }
True: {8, 10, 12, 14, 16, 18, . . .}
{a− b : a, b ∈ N} =
Z
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Set-Builder Notation
A︸︷︷︸name
= { expression︸ ︷︷ ︸all things of the form
: rule︸︷︷︸such that
}
Examples:
{n : n is prime} (set of all primes)
E = {2n : n ∈ N}= {2(1), 2(2), 2(3), 2(4), . . .}= {2, 4, 6, 8, . . .}Positive even numbers
{n ∈ N :√n ∈ N} set of all perfect squares
Set of all odd numbers:
{2n + 1 : n ∈ Z}
All integers not divisible by 3:
{n ∈ Z :
n
36∈ Z
}
True or false: the following sets are equal.
{ −2n : n ∈ Z and −2n > 7 }{ 2(n + 1) : n ∈ N and n > 2 }{ 2n : n ∈ Z and n ≥ 4 }
True: {8, 10, 12, 14, 16, 18, . . .}
{a− b : a, b ∈ N} =
Z
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Set-Builder Notation
A︸︷︷︸name
= { expression︸ ︷︷ ︸all things of the form
: rule︸︷︷︸such that
}
Examples:
{n : n is prime} (set of all primes)
E = {2n : n ∈ N}= {2(1), 2(2), 2(3), 2(4), . . .}= {2, 4, 6, 8, . . .}Positive even numbers
{n ∈ N :√n ∈ N} set of all perfect squares
Set of all odd numbers: {2n + 1 : n ∈ Z}
All integers not divisible by 3:
{n ∈ Z :
n
36∈ Z
}
True or false: the following sets are equal.
{ −2n : n ∈ Z and −2n > 7 }{ 2(n + 1) : n ∈ N and n > 2 }{ 2n : n ∈ Z and n ≥ 4 }
True: {8, 10, 12, 14, 16, 18, . . .}
{a− b : a, b ∈ N} =
Z
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Set-Builder Notation
A︸︷︷︸name
= { expression︸ ︷︷ ︸all things of the form
: rule︸︷︷︸such that
}
Examples:
{n : n is prime} (set of all primes)
E = {2n : n ∈ N}= {2(1), 2(2), 2(3), 2(4), . . .}= {2, 4, 6, 8, . . .}Positive even numbers
{n ∈ N :√n ∈ N} set of all perfect squares
Set of all odd numbers: {2n + 1 : n ∈ Z}
All integers not divisible by 3:{n ∈ Z :
n
36∈ Z
}True or false: the following sets are equal.
{ −2n : n ∈ Z and −2n > 7 }{ 2(n + 1) : n ∈ N and n > 2 }{ 2n : n ∈ Z and n ≥ 4 }
True: {8, 10, 12, 14, 16, 18, . . .}
{a− b : a, b ∈ N} =
Z
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Set-Builder Notation
A︸︷︷︸name
= { expression︸ ︷︷ ︸all things of the form
: rule︸︷︷︸such that
}
Examples:
{n : n is prime} (set of all primes)
E = {2n : n ∈ N}= {2(1), 2(2), 2(3), 2(4), . . .}= {2, 4, 6, 8, . . .}Positive even numbers
{n ∈ N :√n ∈ N} set of all perfect squares
Set of all odd numbers: {2n + 1 : n ∈ Z}
All integers not divisible by 3:{n ∈ Z :
n
36∈ Z
}True or false: the following sets are equal.
{ −2n : n ∈ Z and −2n > 7 }{ 2(n + 1) : n ∈ N and n > 2 }{ 2n : n ∈ Z and n ≥ 4 }
True: {8, 10, 12, 14, 16, 18, . . .}{a− b : a, b ∈ N} =
Z
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Set-Builder Notation
A︸︷︷︸name
= { expression︸ ︷︷ ︸all things of the form
: rule︸︷︷︸such that
}
Examples:
{n : n is prime} (set of all primes)
E = {2n : n ∈ N}= {2(1), 2(2), 2(3), 2(4), . . .}= {2, 4, 6, 8, . . .}Positive even numbers
{n ∈ N :√n ∈ N} set of all perfect squares
Set of all odd numbers: {2n + 1 : n ∈ Z}
All integers not divisible by 3:{n ∈ Z :
n
36∈ Z
}True or false: the following sets are equal.
{ −2n : n ∈ Z and −2n > 7 }{ 2(n + 1) : n ∈ N and n > 2 }{ 2n : n ∈ Z and n ≥ 4 }
True: {8, 10, 12, 14, 16, 18, . . .}{a− b : a, b ∈ N} = Z
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
More Special Sets
Q ={mn
: m, n ∈ Z and n 6= 0}
Rational Numbers (ratios of integers)
R: Real numbers (set of all numbers on the number line)[a, b] = {x ∈ R : a ≤ x ≤ b} closed interval(a, b) =
{x ∈ R : a < x < b}
open interval[a, b) =
{x ∈ R : a ≤ x < b}
half-closed interval[a,∞) =
{x ∈ R : x ≥ a}
infinite interval
Exercises: On the xy -plane, sketch the points in the sets below.
{(x , y) : x = y}{(x , y) : |x | < 1, |y | < 2}{(2x , y) : x < y}{(x , y) : x , y ∈ N}{(x , x + y) : x ∈ R, y ∈ Z}
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
More Special Sets
Q ={mn
: m, n ∈ Z and n 6= 0}
Rational Numbers (ratios of integers)
R: Real numbers (set of all numbers on the number line)[a, b] = {x ∈ R : a ≤ x ≤ b} closed interval(a, b) =
{x ∈ R : a < x < b}
open interval[a, b) =
{x ∈ R : a ≤ x < b}
half-closed interval[a,∞) =
{x ∈ R : x ≥ a}
infinite interval
Exercises: On the xy -plane, sketch the points in the sets below.
{(x , y) : x = y}{(x , y) : |x | < 1, |y | < 2}{(2x , y) : x < y}{(x , y) : x , y ∈ N}{(x , x + y) : x ∈ R, y ∈ Z}
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
More Special Sets
Q ={mn
: m, n ∈ Z and n 6= 0}
Rational Numbers (ratios of integers)
R: Real numbers (set of all numbers on the number line)
[a, b] = {x ∈ R : a ≤ x ≤ b} closed interval(a, b) =
{x ∈ R : a < x < b}
open interval[a, b) =
{x ∈ R : a ≤ x < b}
half-closed interval[a,∞) =
{x ∈ R : x ≥ a}
infinite interval
Exercises: On the xy -plane, sketch the points in the sets below.
{(x , y) : x = y}{(x , y) : |x | < 1, |y | < 2}{(2x , y) : x < y}{(x , y) : x , y ∈ N}{(x , x + y) : x ∈ R, y ∈ Z}
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
More Special Sets
Q ={mn
: m, n ∈ Z and n 6= 0}
Rational Numbers (ratios of integers)
R: Real numbers (set of all numbers on the number line)[a, b] = {x ∈ R : a ≤ x ≤ b}
closed interval(a, b) =
{x ∈ R : a < x < b}
open interval[a, b) =
{x ∈ R : a ≤ x < b}
half-closed interval[a,∞) =
{x ∈ R : x ≥ a}
infinite interval
Exercises: On the xy -plane, sketch the points in the sets below.
{(x , y) : x = y}{(x , y) : |x | < 1, |y | < 2}{(2x , y) : x < y}{(x , y) : x , y ∈ N}{(x , x + y) : x ∈ R, y ∈ Z}
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
More Special Sets
Q ={mn
: m, n ∈ Z and n 6= 0}
Rational Numbers (ratios of integers)
R: Real numbers (set of all numbers on the number line)[a, b] = {x ∈ R : a ≤ x ≤ b} closed interval
(a, b) =
{x ∈ R : a < x < b}
open interval[a, b) =
{x ∈ R : a ≤ x < b}
half-closed interval[a,∞) =
{x ∈ R : x ≥ a}
infinite interval
Exercises: On the xy -plane, sketch the points in the sets below.
{(x , y) : x = y}{(x , y) : |x | < 1, |y | < 2}{(2x , y) : x < y}{(x , y) : x , y ∈ N}{(x , x + y) : x ∈ R, y ∈ Z}
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
More Special Sets
Q ={mn
: m, n ∈ Z and n 6= 0}
Rational Numbers (ratios of integers)
R: Real numbers (set of all numbers on the number line)[a, b] = {x ∈ R : a ≤ x ≤ b} closed interval(a, b) =
{x ∈ R : a < x < b}
open interval[a, b) =
{x ∈ R : a ≤ x < b}
half-closed interval[a,∞) =
{x ∈ R : x ≥ a}
infinite interval
Exercises: On the xy -plane, sketch the points in the sets below.
{(x , y) : x = y}{(x , y) : |x | < 1, |y | < 2}{(2x , y) : x < y}{(x , y) : x , y ∈ N}{(x , x + y) : x ∈ R, y ∈ Z}
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
More Special Sets
Q ={mn
: m, n ∈ Z and n 6= 0}
Rational Numbers (ratios of integers)
R: Real numbers (set of all numbers on the number line)[a, b] = {x ∈ R : a ≤ x ≤ b} closed interval(a, b) ={x ∈ R : a < x < b} open interval[a, b) =
{x ∈ R : a ≤ x < b}
half-closed interval[a,∞) =
{x ∈ R : x ≥ a}
infinite interval
Exercises: On the xy -plane, sketch the points in the sets below.
{(x , y) : x = y}{(x , y) : |x | < 1, |y | < 2}{(2x , y) : x < y}{(x , y) : x , y ∈ N}{(x , x + y) : x ∈ R, y ∈ Z}
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
More Special Sets
Q ={mn
: m, n ∈ Z and n 6= 0}
Rational Numbers (ratios of integers)
R: Real numbers (set of all numbers on the number line)[a, b] = {x ∈ R : a ≤ x ≤ b} closed interval(a, b) ={x ∈ R : a < x < b} open interval[a, b) ={x ∈ R : a ≤ x < b} half-closed interval[a,∞) =
{x ∈ R : x ≥ a}
infinite interval
Exercises: On the xy -plane, sketch the points in the sets below.
{(x , y) : x = y}{(x , y) : |x | < 1, |y | < 2}{(2x , y) : x < y}{(x , y) : x , y ∈ N}{(x , x + y) : x ∈ R, y ∈ Z}
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
More Special Sets
Q ={mn
: m, n ∈ Z and n 6= 0}
Rational Numbers (ratios of integers)
R: Real numbers (set of all numbers on the number line)[a, b] = {x ∈ R : a ≤ x ≤ b} closed interval(a, b) ={x ∈ R : a < x < b} open interval[a, b) ={x ∈ R : a ≤ x < b} half-closed interval[a,∞) ={x ∈ R : x ≥ a} infinite interval
Exercises: On the xy -plane, sketch the points in the sets below.
{(x , y) : x = y}{(x , y) : |x | < 1, |y | < 2}{(2x , y) : x < y}{(x , y) : x , y ∈ N}{(x , x + y) : x ∈ R, y ∈ Z}
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
More Special Sets
Q ={mn
: m, n ∈ Z and n 6= 0}
Rational Numbers (ratios of integers)
R: Real numbers (set of all numbers on the number line)[a, b] = {x ∈ R : a ≤ x ≤ b} closed interval(a, b) ={x ∈ R : a < x < b} open interval[a, b) ={x ∈ R : a ≤ x < b} half-closed interval[a,∞) ={x ∈ R : x ≥ a} infinite interval
Exercises: On the xy -plane, sketch the points in the sets below.
{(x , y) : x = y}{(x , y) : |x | < 1, |y | < 2}{(2x , y) : x < y}{(x , y) : x , y ∈ N}{(x , x + y) : x ∈ R, y ∈ Z}
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Motivation and Definition
Suppose we have sets L and T , where L is a set of locations and T is a set oftimes. We want to pair them, so that we have a new set of elements that tellsus both time and location.
Definition
An ordered pair is a list (x , y) of two things (x and y) such that
(x , y) = (a, b) if and only if x = a AND y = b. That is: order matters.
You have seen these used to describe points in the xy -plane.
Definition
The Cartesian product of two sets A and B is another set, denoted as
A× B (“A cross B”) and defined as
A× B = {(a, b) : a ∈ A and b ∈ B}.
So, T × L is a set, whose elements have the form (t, l):ordered pairs of times and locations.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Motivation and Definition
Suppose we have sets L and T , where L is a set of locations and T is a set oftimes. We want to pair them, so that we have a new set of elements that tellsus both time and location.
Definition
An ordered pair is a list (x , y) of two things (x and y) such that
(x , y) = (a, b) if and only if x = a AND y = b. That is: order matters.
You have seen these used to describe points in the xy -plane.
Definition
The Cartesian product of two sets A and B is another set, denoted as
A× B (“A cross B”) and defined as
A× B = {(a, b) : a ∈ A and b ∈ B}.
So, T × L is a set, whose elements have the form (t, l):ordered pairs of times and locations.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Cartesian Product
We can visualize the pairs as a grid, similar to the xy -plane.Suppose T = {noon, 4pm, 8pm} andL = {in a box, with a fox, in a house, with a mouse}.
no
on
4p
m
8p
m
box
fox
house
mouse
(4pm,box) (8pm,box)(noon,box)
(4pm,fox) (8pm,fox)(noon,fox)
(4pm,house) (8pm,house)(noon,house)
(4pm,mouse) (8pm,mouse)(noon,mouse)
T
L
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Cartesian Products
Theorem
Suppose A and B are finite sets. Then
|A× B| =
|A||B|
Sketch the following sets:
R× {−1, 1}
x
y
Z× N
x
y
Z× R
x
y
The xy -plane is sometimes called R2, since it is R× R.You might have also seen xyz-coordinates called R3.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Cartesian Products
Theorem
Suppose A and B are finite sets. Then
|A× B| = |A||B|
Sketch the following sets:
R× {−1, 1}
x
y
Z× N
x
y
Z× R
x
y
The xy -plane is sometimes called R2, since it is R× R.You might have also seen xyz-coordinates called R3.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Cartesian Products
Theorem
Suppose A and B are finite sets. Then
|A× B| = |A||B|
Sketch the following sets:
R× {−1, 1}
x
y
Z× N
x
y
Z× R
x
y
The xy -plane is sometimes called R2, since it is R× R.
You might have also seen xyz-coordinates called R3.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Cartesian Products
Theorem
Suppose A and B are finite sets. Then
|A× B| = |A||B|
Sketch the following sets:
R× {−1, 1}
x
y
Z× N
x
y
Z× R
x
y
The xy -plane is sometimes called R2, since it is R× R.You might have also seen xyz-coordinates called R3.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Cartesian Products
Theorem
Suppose A and B are finite sets. Then
|A× B| = |A||B|
Sketch the following sets:
R× {−1, 1}
x
y
Z× N
x
y
Z× R
x
y
The xy -plane is sometimes called R2, since it is R× R.You might have also seen xyz-coordinates called R3.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Cartesian Products
Theorem
Suppose A and B are finite sets. Then
|A× B| = |A||B|
Sketch the following sets:
R× {−1, 1}
x
y
Z× N
x
y
Z× R
x
y
The xy -plane is sometimes called R2, since it is R× R.You might have also seen xyz-coordinates called R3.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Higher Orders
Definition
An ordered triple is a list of three things, such as (x , y , z).
(a, b, c)
We take the Cartesian product of n sets A1, A2, . . . , An in exactly the wayyou’d imagine:
A1 × A2 × · · · × An = {(a1, a2, . . . , an) : ai ∈ Ai for all i = 1, 2, . . . , n}
We write An = A× A× · · · × A︸ ︷︷ ︸n times
Example
The dimensions of a cardboard box can be written as (x , y , z) ∈ R3.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Higher Orders
Definition
An ordered triple is a list of three things, such as (x , y , z).
(a, b, c)
We take the Cartesian product of n sets A1, A2, . . . , An in exactly the wayyou’d imagine:
A1 × A2 × · · · × An = {(a1, a2, . . . , an) : ai ∈ Ai for all i = 1, 2, . . . , n}
We write An = A× A× · · · × A︸ ︷︷ ︸n times
Example
The dimensions of a cardboard box can be written as (x , y , z) ∈ R3.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Higher Orders
Definition
An ordered triple is a list of three things, such as (x , y , z).
(a, b, c)
We take the Cartesian product of n sets A1, A2, . . . , An in exactly the wayyou’d imagine:
A1 × A2 × · · · × An = {(a1, a2, . . . , an) : ai ∈ Ai for all i = 1, 2, . . . , n}
We write An = A× A× · · · × A︸ ︷︷ ︸n times
Example
The dimensions of a cardboard box can be written as (x , y , z) ∈ R3.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Higher Orders
Definition
An ordered triple is a list of three things, such as (x , y , z).
(a, b, c)
We take the Cartesian product of n sets A1, A2, . . . , An in exactly the wayyou’d imagine:
A1 × A2 × · · · × An = {(a1, a2, . . . , an) : ai ∈ Ai for all i = 1, 2, . . . , n}
We write An = A× A× · · · × A︸ ︷︷ ︸n times
Example
The dimensions of a cardboard box can be written as (x , y , z) ∈ R3.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Higher Orders
Example
Let A = {1, 2}, B = {3}, and C = {1, 3}. Then
A× B × C =
{(1, 3, 1), (1, 3, 3), (2, 3, 1), (2, 3, 3)}
Note |A× B × C | = 4.
Another example:Let A = {1, 2, 3}, B = {x , y}, C = {A,B}, and D = {10, 20, 30}.What is |A× B × C × D|?
(3)(2)(2)(3) = 36
Theorem
|A1 × A2 × · · · × An| =
|A1||A2| · · · |An|
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Higher Orders
Example
Let A = {1, 2}, B = {3}, and C = {1, 3}. Then
A× B × C = {(1, 3, 1), (1, 3, 3), (2, 3, 1), (2, 3, 3)}
Note |A× B × C | = 4.
Another example:Let A = {1, 2, 3}, B = {x , y}, C = {A,B}, and D = {10, 20, 30}.What is |A× B × C × D|?
(3)(2)(2)(3) = 36
Theorem
|A1 × A2 × · · · × An| =
|A1||A2| · · · |An|
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Higher Orders
Example
Let A = {1, 2}, B = {3}, and C = {1, 3}. Then
A× B × C = {(1, 3, 1), (1, 3, 3), (2, 3, 1), (2, 3, 3)}
Note |A× B × C | = 4.
Another example:Let A = {1, 2, 3}, B = {x , y}, C = {A,B}, and D = {10, 20, 30}.What is |A× B × C × D|?
(3)(2)(2)(3) = 36
Theorem
|A1 × A2 × · · · × An| =
|A1||A2| · · · |An|
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Higher Orders
Example
Let A = {1, 2}, B = {3}, and C = {1, 3}. Then
A× B × C = {(1, 3, 1), (1, 3, 3), (2, 3, 1), (2, 3, 3)}
Note |A× B × C | = 4.
Another example:Let A = {1, 2, 3}, B = {x , y}, C = {A,B}, and D = {10, 20, 30}.What is |A× B × C × D|?
(3)(2)(2)(3) = 36
Theorem
|A1 × A2 × · · · × An| =
|A1||A2| · · · |An|
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Higher Orders
Example
Let A = {1, 2}, B = {3}, and C = {1, 3}. Then
A× B × C = {(1, 3, 1), (1, 3, 3), (2, 3, 1), (2, 3, 3)}
Note |A× B × C | = 4.
Another example:Let A = {1, 2, 3}, B = {x , y}, C = {A,B}, and D = {10, 20, 30}.What is |A× B × C × D|?
(3)(2)(2)(3) = 36
{(a, b, c, d) : a ∈ A, b ∈ B, c ∈ C , d ∈ D}
(1, (2, (3,
(1,x (1,y (2,x (2,y (3,x (3,y
(1,x,A, (1,x,B, (1,y,A, (1,y,B, (2,x,A, (2,x,B, (2,y,A, (2,y,B, (3,x,A, (3,x,B, (3,y,A, (3,y,B,
(1,x,A,30)(1,x,A,20)(1,x,A,10)
(1,x,B,30)(1,x,B,20)(1,x,B,10)
(1,y,A,30)(1,y,A,20)(1,y,A,10)
(1,y,B,30)(1,y,B,20)(1,y,B,10)
(2,x,A,30)(2,x,A,20)(2,x,A,10)
(2,x,B,30)(2,x,B,20)(2,x,B,10)
(2,y,A,30)(2,y,A,20)(2,y,A,10)
(2,y,B,30)(2,y,B,20)(2,y,B,10)
(3,x,A,30)(3,x,A,20)(3,x,A,10)
(3,x,B,30)(3,x,B,20)(3,x,B,10)
(3,y,A,30)(3,y,A,20)(3,y,A,10)
(3,y,B,30)(3,y,B,20)(3,y,B,10)
Theorem
|A1 × A2 × · · · × An| =
|A1||A2| · · · |An|
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Higher Orders
Example
Let A = {1, 2}, B = {3}, and C = {1, 3}. Then
A× B × C = {(1, 3, 1), (1, 3, 3), (2, 3, 1), (2, 3, 3)}
Note |A× B × C | = 4.
Another example:Let A = {1, 2, 3}, B = {x , y}, C = {A,B}, and D = {10, 20, 30}.What is |A× B × C × D|?
(3)(2)(2)(3) = 36
{(a, b, c, d) : a ∈ A, b ∈ B, c ∈ C , d ∈ D}(1, (2, (3,
(1,x (1,y (2,x (2,y (3,x (3,y
(1,x,A, (1,x,B, (1,y,A, (1,y,B, (2,x,A, (2,x,B, (2,y,A, (2,y,B, (3,x,A, (3,x,B, (3,y,A, (3,y,B,
(1,x,A,30)(1,x,A,20)(1,x,A,10)
(1,x,B,30)(1,x,B,20)(1,x,B,10)
(1,y,A,30)(1,y,A,20)(1,y,A,10)
(1,y,B,30)(1,y,B,20)(1,y,B,10)
(2,x,A,30)(2,x,A,20)(2,x,A,10)
(2,x,B,30)(2,x,B,20)(2,x,B,10)
(2,y,A,30)(2,y,A,20)(2,y,A,10)
(2,y,B,30)(2,y,B,20)(2,y,B,10)
(3,x,A,30)(3,x,A,20)(3,x,A,10)
(3,x,B,30)(3,x,B,20)(3,x,B,10)
(3,y,A,30)(3,y,A,20)(3,y,A,10)
(3,y,B,30)(3,y,B,20)(3,y,B,10)
Theorem
|A1 × A2 × · · · × An| =
|A1||A2| · · · |An|
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Higher Orders
Example
Let A = {1, 2}, B = {3}, and C = {1, 3}. Then
A× B × C = {(1, 3, 1), (1, 3, 3), (2, 3, 1), (2, 3, 3)}
Note |A× B × C | = 4.
Another example:Let A = {1, 2, 3}, B = {x , y}, C = {A,B}, and D = {10, 20, 30}.What is |A× B × C × D|?
(3)(2)(2)(3) = 36
{(a, b, c, d) : a ∈ A, b ∈ B, c ∈ C , d ∈ D}(1, (2, (3,
(1,x (1,y (2,x (2,y (3,x (3,y
(1,x,A, (1,x,B, (1,y,A, (1,y,B, (2,x,A, (2,x,B, (2,y,A, (2,y,B, (3,x,A, (3,x,B, (3,y,A, (3,y,B,
(1,x,A,30)(1,x,A,20)(1,x,A,10)
(1,x,B,30)(1,x,B,20)(1,x,B,10)
(1,y,A,30)(1,y,A,20)(1,y,A,10)
(1,y,B,30)(1,y,B,20)(1,y,B,10)
(2,x,A,30)(2,x,A,20)(2,x,A,10)
(2,x,B,30)(2,x,B,20)(2,x,B,10)
(2,y,A,30)(2,y,A,20)(2,y,A,10)
(2,y,B,30)(2,y,B,20)(2,y,B,10)
(3,x,A,30)(3,x,A,20)(3,x,A,10)
(3,x,B,30)(3,x,B,20)(3,x,B,10)
(3,y,A,30)(3,y,A,20)(3,y,A,10)
(3,y,B,30)(3,y,B,20)(3,y,B,10)
Theorem
|A1 × A2 × · · · × An| =
|A1||A2| · · · |An|
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Higher Orders
Example
Let A = {1, 2}, B = {3}, and C = {1, 3}. Then
A× B × C = {(1, 3, 1), (1, 3, 3), (2, 3, 1), (2, 3, 3)}
Note |A× B × C | = 4.
Another example:Let A = {1, 2, 3}, B = {x , y}, C = {A,B}, and D = {10, 20, 30}.What is |A× B × C × D|?
(3)(2)(2)(3) = 36
{(a, b, c, d) : a ∈ A, b ∈ B, c ∈ C , d ∈ D}(1, (2, (3,
(1,x (1,y (2,x (2,y (3,x (3,y
(1,x,A, (1,x,B, (1,y,A, (1,y,B, (2,x,A, (2,x,B, (2,y,A, (2,y,B, (3,x,A, (3,x,B, (3,y,A, (3,y,B,
(1,x,A,30)(1,x,A,20)(1,x,A,10)
(1,x,B,30)(1,x,B,20)(1,x,B,10)
(1,y,A,30)(1,y,A,20)(1,y,A,10)
(1,y,B,30)(1,y,B,20)(1,y,B,10)
(2,x,A,30)(2,x,A,20)(2,x,A,10)
(2,x,B,30)(2,x,B,20)(2,x,B,10)
(2,y,A,30)(2,y,A,20)(2,y,A,10)
(2,y,B,30)(2,y,B,20)(2,y,B,10)
(3,x,A,30)(3,x,A,20)(3,x,A,10)
(3,x,B,30)(3,x,B,20)(3,x,B,10)
(3,y,A,30)(3,y,A,20)(3,y,A,10)
(3,y,B,30)(3,y,B,20)(3,y,B,10)
Theorem
|A1 × A2 × · · · × An| =
|A1||A2| · · · |An|
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Higher Orders
Example
Let A = {1, 2}, B = {3}, and C = {1, 3}. Then
A× B × C = {(1, 3, 1), (1, 3, 3), (2, 3, 1), (2, 3, 3)}
Note |A× B × C | = 4.
Another example:Let A = {1, 2, 3}, B = {x , y}, C = {A,B}, and D = {10, 20, 30}.What is |A× B × C × D|?
(3)(2)(2)(3) = 36
{(a, b, c, d) : a ∈ A, b ∈ B, c ∈ C , d ∈ D}(1, (2, (3,
(1,x (1,y (2,x (2,y (3,x (3,y
(1,x,A, (1,x,B, (1,y,A, (1,y,B, (2,x,A, (2,x,B, (2,y,A, (2,y,B, (3,x,A, (3,x,B, (3,y,A, (3,y,B,
(1,x,A,30)(1,x,A,20)(1,x,A,10)
(1,x,B,30)(1,x,B,20)(1,x,B,10)
(1,y,A,30)(1,y,A,20)(1,y,A,10)
(1,y,B,30)(1,y,B,20)(1,y,B,10)
(2,x,A,30)(2,x,A,20)(2,x,A,10)
(2,x,B,30)(2,x,B,20)(2,x,B,10)
(2,y,A,30)(2,y,A,20)(2,y,A,10)
(2,y,B,30)(2,y,B,20)(2,y,B,10)
(3,x,A,30)(3,x,A,20)(3,x,A,10)
(3,x,B,30)(3,x,B,20)(3,x,B,10)
(3,y,A,30)(3,y,A,20)(3,y,A,10)
(3,y,B,30)(3,y,B,20)(3,y,B,10)
Theorem
|A1 × A2 × · · · × An| =
|A1||A2| · · · |An|
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Higher Orders
Example
Let A = {1, 2}, B = {3}, and C = {1, 3}. Then
A× B × C = {(1, 3, 1), (1, 3, 3), (2, 3, 1), (2, 3, 3)}
Note |A× B × C | = 4.
Another example:Let A = {1, 2, 3}, B = {x , y}, C = {A,B}, and D = {10, 20, 30}.What is |A× B × C × D|? (3)(2)(2)(3) = 36
{(a, b, c, d) : a ∈ A, b ∈ B, c ∈ C , d ∈ D}(1, (2, (3,
(1,x (1,y (2,x (2,y (3,x (3,y
(1,x,A, (1,x,B, (1,y,A, (1,y,B, (2,x,A, (2,x,B, (2,y,A, (2,y,B, (3,x,A, (3,x,B, (3,y,A, (3,y,B,
(1,x,A,30)(1,x,A,20)(1,x,A,10)
(1,x,B,30)(1,x,B,20)(1,x,B,10)
(1,y,A,30)(1,y,A,20)(1,y,A,10)
(1,y,B,30)(1,y,B,20)(1,y,B,10)
(2,x,A,30)(2,x,A,20)(2,x,A,10)
(2,x,B,30)(2,x,B,20)(2,x,B,10)
(2,y,A,30)(2,y,A,20)(2,y,A,10)
(2,y,B,30)(2,y,B,20)(2,y,B,10)
(3,x,A,30)(3,x,A,20)(3,x,A,10)
(3,x,B,30)(3,x,B,20)(3,x,B,10)
(3,y,A,30)(3,y,A,20)(3,y,A,10)
(3,y,B,30)(3,y,B,20)(3,y,B,10)
Theorem
|A1 × A2 × · · · × An| =
|A1||A2| · · · |An|
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Higher Orders
Example
Let A = {1, 2}, B = {3}, and C = {1, 3}. Then
A× B × C = {(1, 3, 1), (1, 3, 3), (2, 3, 1), (2, 3, 3)}
Note |A× B × C | = 4.
Another example:Let A = {1, 2, 3}, B = {x , y}, C = {A,B}, and D = {10, 20, 30}.What is |A× B × C × D|? (3)(2)(2)(3) = 36
Theorem
|A1 × A2 × · · · × An| =
|A1||A2| · · · |An|
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Higher Orders
Example
Let A = {1, 2}, B = {3}, and C = {1, 3}. Then
A× B × C = {(1, 3, 1), (1, 3, 3), (2, 3, 1), (2, 3, 3)}
Note |A× B × C | = 4.
Another example:Let A = {1, 2, 3}, B = {x , y}, C = {A,B}, and D = {10, 20, 30}.What is |A× B × C × D|? (3)(2)(2)(3) = 36
Theorem
|A1 × A2 × · · · × An| = |A1||A2| · · · |An|
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Higher Orders
What shape does [−1, 1]3 make?
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Higher Orders
What shape does [−1, 1]3 make? [−1, 1]3 = [−1, 1]︸ ︷︷ ︸x
× [−1, 1]︸ ︷︷ ︸y
× [−1, 1]︸ ︷︷ ︸z
x
y
z
x
y
z
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Higher Orders
What shape does [−1, 1]3 make? [−1, 1]3 = [−1, 1]︸ ︷︷ ︸x
× [−1, 1]︸ ︷︷ ︸y
× [−1, 1]︸ ︷︷ ︸z
x
y
z
x
y
z
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Higher Orders
What shape does [−1, 1]3 make? [−1, 1]3 = [−1, 1]︸ ︷︷ ︸x
× [−1, 1]︸ ︷︷ ︸y
× [−1, 1]︸ ︷︷ ︸z
x
y
z
x
y
z
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Higher Orders
What shape does [−1, 1]3 make? [−1, 1]3 = [−1, 1]︸ ︷︷ ︸x
× [−1, 1]︸ ︷︷ ︸y
× [−1, 1]︸ ︷︷ ︸z
x
y
z
x
y
z
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Higher Orders
What shape does [−1, 1]3 make? [−1, 1]3 = [−1, 1]︸ ︷︷ ︸x
× [−1, 1]︸ ︷︷ ︸y
× [−1, 1]︸ ︷︷ ︸z
x
y
z
x
y
z
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Higher Orders
What shape does [−1, 1]3 make? [−1, 1]3 = [−1, 1]︸ ︷︷ ︸x
× [−1, 1]︸ ︷︷ ︸y
× [−1, 1]︸ ︷︷ ︸z
x
y
z
x
y
z
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definition and First Examples
Definition
A set A is a subset of a set B is every element of A is also an element of B.We write
A ⊆ B
If A is not a subset of B, we write
A 6⊆ B
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
A ⊆ B C ⊆ A,B True or false: A ⊆ A True
True or false: if X and Y are finite sets, and X ⊆ Y , then |X | < |Y |.
False (see next question)
If X and Y are finite sets, X ⊆ Y and |X | ≥ |Y |, what do you know aboutthe relationship between X and Y ?
X = Y (and so |X | = |Y |)
True or false: for any set X , ∅ ⊆ X .
True
If X ⊆ Y and Y ⊆ X , what do you know about X and Y ?
X = Y (!!!)
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definition and First Examples
Definition
A set A is a subset of a set B is every element of A is also an element of B.We write
A ⊆ B
If A is not a subset of B, we write
A 6⊆ B
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
A ⊆ B C ⊆ A,B True or false: A ⊆ A True
True or false: if X and Y are finite sets, and X ⊆ Y , then |X | < |Y |.
False (see next question)
If X and Y are finite sets, X ⊆ Y and |X | ≥ |Y |, what do you know aboutthe relationship between X and Y ?
X = Y (and so |X | = |Y |)
True or false: for any set X , ∅ ⊆ X .
True
If X ⊆ Y and Y ⊆ X , what do you know about X and Y ?
X = Y (!!!)
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definition and First Examples
Definition
A set A is a subset of a set B is every element of A is also an element of B.We write
A ⊆ B
If A is not a subset of B, we write
A 6⊆ B
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
A ⊆ B C ⊆ A,B True or false: A ⊆ A True
True or false: if X and Y are finite sets, and X ⊆ Y , then |X | < |Y |.
False (see next question)
If X and Y are finite sets, X ⊆ Y and |X | ≥ |Y |, what do you know aboutthe relationship between X and Y ?
X = Y (and so |X | = |Y |)
True or false: for any set X , ∅ ⊆ X .
True
If X ⊆ Y and Y ⊆ X , what do you know about X and Y ?
X = Y (!!!)
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definition and First Examples
Definition
A set A is a subset of a set B is every element of A is also an element of B.We write
A ⊆ B
If A is not a subset of B, we write
A 6⊆ B
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
A ⊆
B C ⊆ A,B True or false: A ⊆ A True
True or false: if X and Y are finite sets, and X ⊆ Y , then |X | < |Y |.
False (see next question)
If X and Y are finite sets, X ⊆ Y and |X | ≥ |Y |, what do you know aboutthe relationship between X and Y ?
X = Y (and so |X | = |Y |)
True or false: for any set X , ∅ ⊆ X .
True
If X ⊆ Y and Y ⊆ X , what do you know about X and Y ?
X = Y (!!!)
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definition and First Examples
Definition
A set A is a subset of a set B is every element of A is also an element of B.We write
A ⊆ B
If A is not a subset of B, we write
A 6⊆ B
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
A ⊆ B
C ⊆ A,B True or false: A ⊆ A True
True or false: if X and Y are finite sets, and X ⊆ Y , then |X | < |Y |.
False (see next question)
If X and Y are finite sets, X ⊆ Y and |X | ≥ |Y |, what do you know aboutthe relationship between X and Y ?
X = Y (and so |X | = |Y |)
True or false: for any set X , ∅ ⊆ X .
True
If X ⊆ Y and Y ⊆ X , what do you know about X and Y ?
X = Y (!!!)
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definition and First Examples
Definition
A set A is a subset of a set B is every element of A is also an element of B.We write
A ⊆ B
If A is not a subset of B, we write
A 6⊆ B
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
A ⊆ B C ⊆
A,B True or false: A ⊆ A True
True or false: if X and Y are finite sets, and X ⊆ Y , then |X | < |Y |.
False (see next question)
If X and Y are finite sets, X ⊆ Y and |X | ≥ |Y |, what do you know aboutthe relationship between X and Y ?
X = Y (and so |X | = |Y |)
True or false: for any set X , ∅ ⊆ X .
True
If X ⊆ Y and Y ⊆ X , what do you know about X and Y ?
X = Y (!!!)
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definition and First Examples
Definition
A set A is a subset of a set B is every element of A is also an element of B.We write
A ⊆ B
If A is not a subset of B, we write
A 6⊆ B
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
A ⊆ B C ⊆ A,B
True or false: A ⊆ A True
True or false: if X and Y are finite sets, and X ⊆ Y , then |X | < |Y |.
False (see next question)
If X and Y are finite sets, X ⊆ Y and |X | ≥ |Y |, what do you know aboutthe relationship between X and Y ?
X = Y (and so |X | = |Y |)
True or false: for any set X , ∅ ⊆ X .
True
If X ⊆ Y and Y ⊆ X , what do you know about X and Y ?
X = Y (!!!)
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definition and First Examples
Definition
A set A is a subset of a set B is every element of A is also an element of B.We write
A ⊆ B
If A is not a subset of B, we write
A 6⊆ B
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
A ⊆ B C ⊆ A,B True or false: A ⊆ A
True
True or false: if X and Y are finite sets, and X ⊆ Y , then |X | < |Y |.
False (see next question)
If X and Y are finite sets, X ⊆ Y and |X | ≥ |Y |, what do you know aboutthe relationship between X and Y ?
X = Y (and so |X | = |Y |)
True or false: for any set X , ∅ ⊆ X .
True
If X ⊆ Y and Y ⊆ X , what do you know about X and Y ?
X = Y (!!!)
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definition and First Examples
Definition
A set A is a subset of a set B is every element of A is also an element of B.We write
A ⊆ B
If A is not a subset of B, we write
A 6⊆ B
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
A ⊆ B C ⊆ A,B True or false: A ⊆ A True
True or false: if X and Y are finite sets, and X ⊆ Y , then |X | < |Y |.
False (see next question)
If X and Y are finite sets, X ⊆ Y and |X | ≥ |Y |, what do you know aboutthe relationship between X and Y ?
X = Y (and so |X | = |Y |)
True or false: for any set X , ∅ ⊆ X .
True
If X ⊆ Y and Y ⊆ X , what do you know about X and Y ?
X = Y (!!!)
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definition and First Examples
Definition
A set A is a subset of a set B is every element of A is also an element of B.We write
A ⊆ B
If A is not a subset of B, we write
A 6⊆ B
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
A ⊆ B C ⊆ A,B True or false: A ⊆ A True
True or false: if X and Y are finite sets, and X ⊆ Y , then |X | < |Y |.
False (see next question)
If X and Y are finite sets, X ⊆ Y and |X | ≥ |Y |, what do you know aboutthe relationship between X and Y ?
X = Y (and so |X | = |Y |)
True or false: for any set X , ∅ ⊆ X .
True
If X ⊆ Y and Y ⊆ X , what do you know about X and Y ?
X = Y (!!!)
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definition and First Examples
Definition
A set A is a subset of a set B is every element of A is also an element of B.We write
A ⊆ B
If A is not a subset of B, we write
A 6⊆ B
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
A ⊆ B C ⊆ A,B True or false: A ⊆ A True
True or false: if X and Y are finite sets, and X ⊆ Y , then |X | < |Y |.
False (see next question)
If X and Y are finite sets, X ⊆ Y and |X | ≥ |Y |, what do you know aboutthe relationship between X and Y ?
X = Y (and so |X | = |Y |)True or false: for any set X , ∅ ⊆ X .
True
If X ⊆ Y and Y ⊆ X , what do you know about X and Y ?
X = Y (!!!)
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definition and First Examples
Definition
A set A is a subset of a set B is every element of A is also an element of B.We write
A ⊆ B
If A is not a subset of B, we write
A 6⊆ B
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
A ⊆ B C ⊆ A,B True or false: A ⊆ A True
True or false: if X and Y are finite sets, and X ⊆ Y , then |X | < |Y |.
False (see next question)
If X and Y are finite sets, X ⊆ Y and |X | ≥ |Y |, what do you know aboutthe relationship between X and Y ?
X = Y (and so |X | = |Y |)
True or false: for any set X , ∅ ⊆ X .
True
If X ⊆ Y and Y ⊆ X , what do you know about X and Y ?
X = Y (!!!)
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definition and First Examples
Definition
A set A is a subset of a set B is every element of A is also an element of B.We write
A ⊆ B
If A is not a subset of B, we write
A 6⊆ B
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
A ⊆ B C ⊆ A,B True or false: A ⊆ A True
True or false: if X and Y are finite sets, and X ⊆ Y , then |X | < |Y |.
False (see next question)
If X and Y are finite sets, X ⊆ Y and |X | ≥ |Y |, what do you know aboutthe relationship between X and Y ?
X = Y (and so |X | = |Y |)
True or false: for any set X , ∅ ⊆ X .
True
If X ⊆ Y and Y ⊆ X , what do you know about X and Y ?
X = Y (!!!)
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definition and First Examples
Definition
A set A is a subset of a set B is every element of A is also an element of B.We write
A ⊆ B
If A is not a subset of B, we write
A 6⊆ B
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
A ⊆ B C ⊆ A,B True or false: A ⊆ A True
True or false: if X and Y are finite sets, and X ⊆ Y , then |X | < |Y |.False (see next question)
If X and Y are finite sets, X ⊆ Y and |X | ≥ |Y |, what do you know aboutthe relationship between X and Y ?
X = Y (and so |X | = |Y |)
True or false: for any set X , ∅ ⊆ X .
True
If X ⊆ Y and Y ⊆ X , what do you know about X and Y ?
X = Y (!!!)
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definition and First Examples
Definition
A set A is a subset of a set B is every element of A is also an element of B.We write
A ⊆ B
If A is not a subset of B, we write
A 6⊆ B
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
A ⊆ B C ⊆ A,B True or false: A ⊆ A True
True or false: if X and Y are finite sets, and X ⊆ Y , then |X | < |Y |.False (see next question)
If X and Y are finite sets, X ⊆ Y and |X | ≥ |Y |, what do you know aboutthe relationship between X and Y ? X = Y (and so |X | = |Y |)True or false: for any set X , ∅ ⊆ X .
True
If X ⊆ Y and Y ⊆ X , what do you know about X and Y ?
X = Y (!!!)
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definition and First Examples
Definition
A set A is a subset of a set B is every element of A is also an element of B.We write
A ⊆ B
If A is not a subset of B, we write
A 6⊆ B
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
A ⊆ B C ⊆ A,B True or false: A ⊆ A True
True or false: if X and Y are finite sets, and X ⊆ Y , then |X | < |Y |.False (see next question)
If X and Y are finite sets, X ⊆ Y and |X | ≥ |Y |, what do you know aboutthe relationship between X and Y ? X = Y (and so |X | = |Y |)True or false: for any set X , ∅ ⊆ X . True
If X ⊆ Y and Y ⊆ X , what do you know about X and Y ?
X = Y (!!!)
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Definition and First Examples
Definition
A set A is a subset of a set B is every element of A is also an element of B.We write
A ⊆ B
If A is not a subset of B, we write
A 6⊆ B
A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}
A ⊆ B C ⊆ A,B True or false: A ⊆ A True
True or false: if X and Y are finite sets, and X ⊆ Y , then |X | < |Y |.False (see next question)
If X and Y are finite sets, X ⊆ Y and |X | ≥ |Y |, what do you know aboutthe relationship between X and Y ? X = Y (and so |X | = |Y |)True or false: for any set X , ∅ ⊆ X . True
If X ⊆ Y and Y ⊆ X , what do you know about X and Y ? X = Y (!!!)
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Notation Review
True or False?
1 ⊆ {1, 2, 3}
False
1 ∈ {1, 2, 3}
True
{1} ⊆ {1, 2, 3}
True
{1} ∈ {1, 2, 3}
False
1 ∈ {{1}, {2, 3}}
False
{1} ∈ {{1}, {2, 3}}
True
1 ⊆ {{1}, {2, 3}}
False
{1} ⊆ {{1}, {2, 3}}
False
{{1}} ⊆ {{1}, {2, 3}}
True
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Notation Review
True or False?
1 ⊆ {1, 2, 3} False
1 ∈ {1, 2, 3} True
{1} ⊆ {1, 2, 3} True
{1} ∈ {1, 2, 3} False
1 ∈ {{1}, {2, 3}} False
{1} ∈ {{1}, {2, 3}} True
1 ⊆ {{1}, {2, 3}} False
{1} ⊆ {{1}, {2, 3}} False
{{1}} ⊆ {{1}, {2, 3}} True
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Listing Subsets
Example: List all subsets of the set S = {1, 2, 3}.
size sets notes0
∅ subset of any set
1
{1} {2} {3}
2
{2, 3} {1, 3} {1, 2} delete one element
3
{1, 2, 3}
Example: List all subsets of the set T = {a, b, {1, 2}}.Note: |T | = 3.
size sets notes0
∅ subset of any set
1
{a} {b} {{1, 2}}
2
{b, {1, 2}} {a, {1, 2}} {a, b} delete one element
3
{a, b, {1, 2}}
How many subsets does any set of cardinality 3 have? 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Listing Subsets
Example: List all subsets of the set S = {1, 2, 3}.
size sets notes0
∅ subset of any set
1
{1} {2} {3}
2
{2, 3} {1, 3} {1, 2} delete one element
3
{1, 2, 3}
Example: List all subsets of the set T = {a, b, {1, 2}}.Note: |T | = 3.
size sets notes0
∅ subset of any set
1
{a} {b} {{1, 2}}
2
{b, {1, 2}} {a, {1, 2}} {a, b} delete one element
3
{a, b, {1, 2}}
How many subsets does any set of cardinality 3 have? 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Listing Subsets
Example: List all subsets of the set S = {1, 2, 3}.
size sets notes0 ∅ subset of any set1
{1} {2} {3}
2
{2, 3} {1, 3} {1, 2} delete one element
3
{1, 2, 3}
Example: List all subsets of the set T = {a, b, {1, 2}}.Note: |T | = 3.
size sets notes0
∅ subset of any set
1
{a} {b} {{1, 2}}
2
{b, {1, 2}} {a, {1, 2}} {a, b} delete one element
3
{a, b, {1, 2}}
How many subsets does any set of cardinality 3 have? 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Listing Subsets
Example: List all subsets of the set S = {1, 2, 3}.
size sets notes0 ∅ subset of any set1 {1} {2} {3}2
{2, 3} {1, 3} {1, 2} delete one element
3
{1, 2, 3}
Example: List all subsets of the set T = {a, b, {1, 2}}.Note: |T | = 3.
size sets notes0
∅ subset of any set
1
{a} {b} {{1, 2}}
2
{b, {1, 2}} {a, {1, 2}} {a, b} delete one element
3
{a, b, {1, 2}}
How many subsets does any set of cardinality 3 have? 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Listing Subsets
Example: List all subsets of the set S = {1, 2, 3}.
size sets notes0 ∅ subset of any set1 {1} {2} {3}2
{2, 3} {1, 3} {1, 2}
delete one element3
{1, 2, 3}
Example: List all subsets of the set T = {a, b, {1, 2}}.Note: |T | = 3.
size sets notes0
∅ subset of any set
1
{a} {b} {{1, 2}}
2
{b, {1, 2}} {a, {1, 2}} {a, b} delete one element
3
{a, b, {1, 2}}
How many subsets does any set of cardinality 3 have? 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Listing Subsets
Example: List all subsets of the set S = {1, 2, 3}.
size sets notes0 ∅ subset of any set1 {1} {2} {3}2 {2, 3}
{1, 3} {1, 2}
delete one element3
{1, 2, 3}
Example: List all subsets of the set T = {a, b, {1, 2}}.Note: |T | = 3.
size sets notes0
∅ subset of any set
1
{a} {b} {{1, 2}}
2
{b, {1, 2}} {a, {1, 2}} {a, b} delete one element
3
{a, b, {1, 2}}
How many subsets does any set of cardinality 3 have? 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Listing Subsets
Example: List all subsets of the set S = {1, 2, 3}.
size sets notes0 ∅ subset of any set1 {1} {2} {3}2 {2, 3} {1, 3}
{1, 2}
delete one element3
{1, 2, 3}
Example: List all subsets of the set T = {a, b, {1, 2}}.Note: |T | = 3.
size sets notes0
∅ subset of any set
1
{a} {b} {{1, 2}}
2
{b, {1, 2}} {a, {1, 2}} {a, b} delete one element
3
{a, b, {1, 2}}
How many subsets does any set of cardinality 3 have? 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Listing Subsets
Example: List all subsets of the set S = {1, 2, 3}.
size sets notes0 ∅ subset of any set1 {1} {2} {3}2 {2, 3} {1, 3} {1, 2} delete one element3
{1, 2, 3}
Example: List all subsets of the set T = {a, b, {1, 2}}.Note: |T | = 3.
size sets notes0
∅ subset of any set
1
{a} {b} {{1, 2}}
2
{b, {1, 2}} {a, {1, 2}} {a, b} delete one element
3
{a, b, {1, 2}}
How many subsets does any set of cardinality 3 have? 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Listing Subsets
Example: List all subsets of the set S = {1, 2, 3}.
size sets notes0 ∅ subset of any set1 {1} {2} {3}2 {2, 3} {1, 3} {1, 2} delete one element3 {1, 2, 3}
Example: List all subsets of the set T = {a, b, {1, 2}}.Note: |T | = 3.
size sets notes0
∅ subset of any set
1
{a} {b} {{1, 2}}
2
{b, {1, 2}} {a, {1, 2}} {a, b} delete one element
3
{a, b, {1, 2}}
How many subsets does any set of cardinality 3 have? 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Listing Subsets
Example: List all subsets of the set S = {1, 2, 3}.
size sets notes0 ∅ subset of any set1 {1} {2} {3}2 {2, 3} {1, 3} {1, 2} delete one element3 {1, 2, 3}
Example: List all subsets of the set T = {a, b, {1, 2}}.
Note: |T | = 3.
size sets notes0
∅ subset of any set
1
{a} {b} {{1, 2}}
2
{b, {1, 2}} {a, {1, 2}} {a, b} delete one element
3
{a, b, {1, 2}}
How many subsets does any set of cardinality 3 have? 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Listing Subsets
Example: List all subsets of the set S = {1, 2, 3}.
size sets notes0 ∅ subset of any set1 {1} {2} {3}2 {2, 3} {1, 3} {1, 2} delete one element3 {1, 2, 3}
Example: List all subsets of the set T = {a, b, {1, 2}}.Note: |T | = 3.
size sets notes0
∅ subset of any set
1
{a} {b} {{1, 2}}
2
{b, {1, 2}} {a, {1, 2}} {a, b} delete one element
3
{a, b, {1, 2}}
How many subsets does any set of cardinality 3 have? 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Listing Subsets
Example: List all subsets of the set S = {1, 2, 3}.
size sets notes0 ∅ subset of any set1 {1} {2} {3}2 {2, 3} {1, 3} {1, 2} delete one element3 {1, 2, 3}
Example: List all subsets of the set T = {a, b, {1, 2}}.Note: |T | = 3.
size sets notes0
∅ subset of any set
1
{a} {b} {{1, 2}}
2
{b, {1, 2}} {a, {1, 2}} {a, b} delete one element
3
{a, b, {1, 2}}
How many subsets does any set of cardinality 3 have? 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Listing Subsets
Example: List all subsets of the set S = {1, 2, 3}.
size sets notes0 ∅ subset of any set1 {1} {2} {3}2 {2, 3} {1, 3} {1, 2} delete one element3 {1, 2, 3}
Example: List all subsets of the set T = {a, b, {1, 2}}.Note: |T | = 3.
size sets notes0 ∅ subset of any set1
{a} {b} {{1, 2}}
2
{b, {1, 2}} {a, {1, 2}} {a, b} delete one element
3
{a, b, {1, 2}}
How many subsets does any set of cardinality 3 have? 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Listing Subsets
Example: List all subsets of the set S = {1, 2, 3}.
size sets notes0 ∅ subset of any set1 {1} {2} {3}2 {2, 3} {1, 3} {1, 2} delete one element3 {1, 2, 3}
Example: List all subsets of the set T = {a, b, {1, 2}}.Note: |T | = 3.
size sets notes0 ∅ subset of any set1 {a} {b} {{1, 2}}2
{b, {1, 2}} {a, {1, 2}} {a, b} delete one element
3
{a, b, {1, 2}}
How many subsets does any set of cardinality 3 have? 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Listing Subsets
Example: List all subsets of the set S = {1, 2, 3}.
size sets notes0 ∅ subset of any set1 {1} {2} {3}2 {2, 3} {1, 3} {1, 2} delete one element3 {1, 2, 3}
Example: List all subsets of the set T = {a, b, {1, 2}}.Note: |T | = 3.
size sets notes0 ∅ subset of any set1 {a} {b} {{1, 2}}2 {b, {1, 2}} {a, {1, 2}} {a, b} delete one element3
{a, b, {1, 2}}
How many subsets does any set of cardinality 3 have? 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Listing Subsets
Example: List all subsets of the set S = {1, 2, 3}.
size sets notes0 ∅ subset of any set1 {1} {2} {3}2 {2, 3} {1, 3} {1, 2} delete one element3 {1, 2, 3}
Example: List all subsets of the set T = {a, b, {1, 2}}.Note: |T | = 3.
size sets notes0 ∅ subset of any set1 {a} {b} {{1, 2}}2 {b, {1, 2}} {a, {1, 2}} {a, b} delete one element3 {a, b, {1, 2}}
How many subsets does any set of cardinality 3 have? 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Listing Subsets
Example: List all subsets of the set S = {1, 2, 3}.
size sets notes0 ∅ subset of any set1 {1} {2} {3}2 {2, 3} {1, 3} {1, 2} delete one element3 {1, 2, 3}
Example: List all subsets of the set T = {a, b, {1, 2}}.Note: |T | = 3.
size sets notes0 ∅ subset of any set1 {a} {b} {{1, 2}}2 {b, {1, 2}} {a, {1, 2}} {a, b} delete one element3 {a, b, {1, 2}}
How many subsets does any set of cardinality 3 have?
8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Listing Subsets
Example: List all subsets of the set S = {1, 2, 3}.
size sets notes0 ∅ subset of any set1 {1} {2} {3}2 {2, 3} {1, 3} {1, 2} delete one element3 {1, 2, 3}
Example: List all subsets of the set T = {a, b, {1, 2}}.Note: |T | = 3.
size sets notes0 ∅ subset of any set1 {a} {b} {{1, 2}}2 {b, {1, 2}} {a, {1, 2}} {a, b} delete one element3 {a, b, {1, 2}}
How many subsets does any set of cardinality 3 have? 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Number of Subsets
Let n be a natural number, or zero. How many subsets does a set ofcardinality n have?
We already saw how to count ordered n-tuples, (a1, a2, . . . , an) with ai ∈ Ai .
S = { 1 , 2 , 3 , 4 , 5 }T = { 2 , 4 , 5 }T = ( 0 , 1 , 0 , 1 , 1 )
U = ( 0 , 0 , 1 , 1 , 0 )U = { 3 , 4 }
Every subset of our set of size n corresponds to some unique ordered n-tuple(with entries from {0, 1}), and every n-tuple corresponds to some uniquesubset.The number of subsets of a set of size n is the same as the size of {0, 1}n(the number of ordered n-tuples where every entry is 0 or 1).
There are 2n subsets of a set of size n.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Number of Subsets
Let n be a natural number, or zero. How many subsets does a set ofcardinality n have?We already saw how to count ordered n-tuples, (a1, a2, . . . , an) with ai ∈ Ai .
S = { 1 , 2 , 3 , 4 , 5 }T = { 2 , 4 , 5 }T = ( 0 , 1 , 0 , 1 , 1 )
U = ( 0 , 0 , 1 , 1 , 0 )U = { 3 , 4 }
Every subset of our set of size n corresponds to some unique ordered n-tuple(with entries from {0, 1}), and every n-tuple corresponds to some uniquesubset.The number of subsets of a set of size n is the same as the size of {0, 1}n(the number of ordered n-tuples where every entry is 0 or 1).
There are 2n subsets of a set of size n.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Number of Subsets
Let n be a natural number, or zero. How many subsets does a set ofcardinality n have?We already saw how to count ordered n-tuples, (a1, a2, . . . , an) with ai ∈ Ai .
S = { 1 , 2 , 3 , 4 , 5 }
T = { 2 , 4 , 5 }T = ( 0 , 1 , 0 , 1 , 1 )
U = ( 0 , 0 , 1 , 1 , 0 )U = { 3 , 4 }
Every subset of our set of size n corresponds to some unique ordered n-tuple(with entries from {0, 1}), and every n-tuple corresponds to some uniquesubset.The number of subsets of a set of size n is the same as the size of {0, 1}n(the number of ordered n-tuples where every entry is 0 or 1).
There are 2n subsets of a set of size n.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Number of Subsets
Let n be a natural number, or zero. How many subsets does a set ofcardinality n have?We already saw how to count ordered n-tuples, (a1, a2, . . . , an) with ai ∈ Ai .
S = { 1 , 2 , 3 , 4 , 5 }T = { 2 , 4 , 5 }
T = ( 0 , 1 , 0 , 1 , 1 )
U = ( 0 , 0 , 1 , 1 , 0 )U = { 3 , 4 }
Every subset of our set of size n corresponds to some unique ordered n-tuple(with entries from {0, 1}), and every n-tuple corresponds to some uniquesubset.The number of subsets of a set of size n is the same as the size of {0, 1}n(the number of ordered n-tuples where every entry is 0 or 1).
There are 2n subsets of a set of size n.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Number of Subsets
Let n be a natural number, or zero. How many subsets does a set ofcardinality n have?We already saw how to count ordered n-tuples, (a1, a2, . . . , an) with ai ∈ Ai .
S = { 1 , 2 , 3 , 4 , 5 }T = { 2 , 4 , 5 }T = ( 0 , 1 , 0 , 1 , 1 )
U = ( 0 , 0 , 1 , 1 , 0 )U = { 3 , 4 }
Every subset of our set of size n corresponds to some unique ordered n-tuple(with entries from {0, 1}), and every n-tuple corresponds to some uniquesubset.The number of subsets of a set of size n is the same as the size of {0, 1}n(the number of ordered n-tuples where every entry is 0 or 1).
There are 2n subsets of a set of size n.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Number of Subsets
Let n be a natural number, or zero. How many subsets does a set ofcardinality n have?We already saw how to count ordered n-tuples, (a1, a2, . . . , an) with ai ∈ Ai .
S = { 1 , 2 , 3 , 4 , 5 }T = { 2 , 4 , 5 }T = ( 0 , 1 , 0 , 1 , 1 )
U = ( 0 , 0 , 1 , 1 , 0 )
U = { 3 , 4 }
Every subset of our set of size n corresponds to some unique ordered n-tuple(with entries from {0, 1}), and every n-tuple corresponds to some uniquesubset.The number of subsets of a set of size n is the same as the size of {0, 1}n(the number of ordered n-tuples where every entry is 0 or 1).
There are 2n subsets of a set of size n.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Number of Subsets
Let n be a natural number, or zero. How many subsets does a set ofcardinality n have?We already saw how to count ordered n-tuples, (a1, a2, . . . , an) with ai ∈ Ai .
S = { 1 , 2 , 3 , 4 , 5 }T = { 2 , 4 , 5 }T = ( 0 , 1 , 0 , 1 , 1 )
U = ( 0 , 0 , 1 , 1 , 0 )U = { 3 , 4 }
Every subset of our set of size n corresponds to some unique ordered n-tuple(with entries from {0, 1}), and every n-tuple corresponds to some uniquesubset.The number of subsets of a set of size n is the same as the size of {0, 1}n(the number of ordered n-tuples where every entry is 0 or 1).
There are 2n subsets of a set of size n.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Number of Subsets
Let n be a natural number, or zero. How many subsets does a set ofcardinality n have?We already saw how to count ordered n-tuples, (a1, a2, . . . , an) with ai ∈ Ai .
S = { 1 , 2 , 3 , 4 , 5 }T = { 2 , 4 , 5 }T = ( 0 , 1 , 0 , 1 , 1 )
U = ( 0 , 0 , 1 , 1 , 0 )U = { 3 , 4 }
Every subset of our set of size n corresponds to some unique ordered n-tuple(with entries from {0, 1}), and every n-tuple corresponds to some uniquesubset.
The number of subsets of a set of size n is the same as the size of {0, 1}n(the number of ordered n-tuples where every entry is 0 or 1).
There are 2n subsets of a set of size n.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Number of Subsets
Let n be a natural number, or zero. How many subsets does a set ofcardinality n have?We already saw how to count ordered n-tuples, (a1, a2, . . . , an) with ai ∈ Ai .
S = { 1 , 2 , 3 , 4 , 5 }T = { 2 , 4 , 5 }T = ( 0 , 1 , 0 , 1 , 1 )
U = ( 0 , 0 , 1 , 1 , 0 )U = { 3 , 4 }
Every subset of our set of size n corresponds to some unique ordered n-tuple(with entries from {0, 1}), and every n-tuple corresponds to some uniquesubset.The number of subsets of a set of size n is the same as the size of {0, 1}n(the number of ordered n-tuples where every entry is 0 or 1).
There are 2n subsets of a set of size n.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Number of Subsets
Let n be a natural number, or zero. How many subsets does a set ofcardinality n have?We already saw how to count ordered n-tuples, (a1, a2, . . . , an) with ai ∈ Ai .
S = { 1 , 2 , 3 , 4 , 5 }T = { 2 , 4 , 5 }T = ( 0 , 1 , 0 , 1 , 1 )
U = ( 0 , 0 , 1 , 1 , 0 )U = { 3 , 4 }
Every subset of our set of size n corresponds to some unique ordered n-tuple(with entries from {0, 1}), and every n-tuple corresponds to some uniquesubset.The number of subsets of a set of size n is the same as the size of {0, 1}n(the number of ordered n-tuples where every entry is 0 or 1).
There are 2n subsets of a set of size n.
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Number of Subsets – Check
There are 2n subsets of a set of size n.
n set subsets 2n
0
∅ ∅
20 = 1
1
{1} ∅, {1}
21 = 2
2
{1, 2} ∅, {1}, {2}, {1, 2}
22 = 4
3
{1, 2, 3} (already done: 8 of them)
23 = 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Number of Subsets – Check
There are 2n subsets of a set of size n.
n set subsets 2n
0
∅ ∅
20 = 1
1
{1} ∅, {1}
21 = 2
2
{1, 2} ∅, {1}, {2}, {1, 2}
22 = 4
3
{1, 2, 3} (already done: 8 of them)
23 = 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Number of Subsets – Check
There are 2n subsets of a set of size n.
n set subsets 2n
0
∅ ∅
20 = 1
1
{1} ∅, {1}
21 = 2
2
{1, 2} ∅, {1}, {2}, {1, 2}
22 = 4
3 {1, 2, 3}
(already done: 8 of them)
23 = 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Number of Subsets – Check
There are 2n subsets of a set of size n.
n set subsets 2n
0
∅ ∅
20 = 1
1
{1} ∅, {1}
21 = 2
2
{1, 2} ∅, {1}, {2}, {1, 2}
22 = 4
3 {1, 2, 3} (already done: 8 of them) 23 = 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Number of Subsets – Check
There are 2n subsets of a set of size n.
n set subsets 2n
0
∅ ∅
20 = 1
1 {1}
∅, {1}
21 = 2
2
{1, 2} ∅, {1}, {2}, {1, 2}
22 = 4
3 {1, 2, 3} (already done: 8 of them) 23 = 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Number of Subsets – Check
There are 2n subsets of a set of size n.
n set subsets 2n
0
∅ ∅
20 = 1
1 {1} ∅, {1} 21 = 2
2
{1, 2} ∅, {1}, {2}, {1, 2}
22 = 4
3 {1, 2, 3} (already done: 8 of them) 23 = 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Number of Subsets – Check
There are 2n subsets of a set of size n.
n set subsets 2n
0
∅ ∅
20 = 1
1 {1} ∅, {1} 21 = 2
2 {1, 2}
∅, {1}, {2}, {1, 2}
22 = 4
3 {1, 2, 3} (already done: 8 of them) 23 = 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Number of Subsets – Check
There are 2n subsets of a set of size n.
n set subsets 2n
0
∅ ∅
20 = 1
1 {1} ∅, {1} 21 = 2
2 {1, 2} ∅, {1}, {2}, {1, 2} 22 = 4
3 {1, 2, 3} (already done: 8 of them) 23 = 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Number of Subsets – Check
There are 2n subsets of a set of size n.
n set subsets 2n
0 ∅
∅
20 = 1
1 {1} ∅, {1} 21 = 2
2 {1, 2} ∅, {1}, {2}, {1, 2} 22 = 4
3 {1, 2, 3} (already done: 8 of them) 23 = 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Number of Subsets – Check
There are 2n subsets of a set of size n.
n set subsets 2n
0 ∅ ∅ 20 = 1
1 {1} ∅, {1} 21 = 2
2 {1, 2} ∅, {1}, {2}, {1, 2} 22 = 4
3 {1, 2, 3} (already done: 8 of them) 23 = 8
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Power Sets
Definition
If A is a set, the power set of A is the set of all subsets of A.
We write P(A) for the power set of A,
P(A) = {X : X ⊆ A}
Example: P ({a, b, c}) ={∅,{a}, {b}, {c},{b, c}, {a, c}, {a, b},{a, b, c}}
True or False: For every set A, A ⊆ P(A)True or False: For every set A, A ∈ P(A)
Theorem
For any finite set A,|P(A)| = 2|A|
There are 193 member nations in the UN. How many ways are there to form acommittee made up of at least two member nations?
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Power Sets
Definition
If A is a set, the power set of A is the set of all subsets of A.
We write P(A) for the power set of A,
P(A) = {X : X ⊆ A}
Example: P ({a, b, c}) =
{∅,{a}, {b}, {c},{b, c}, {a, c}, {a, b},{a, b, c}}
True or False: For every set A, A ⊆ P(A)True or False: For every set A, A ∈ P(A)
Theorem
For any finite set A,|P(A)| = 2|A|
There are 193 member nations in the UN. How many ways are there to form acommittee made up of at least two member nations?
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Power Sets
Definition
If A is a set, the power set of A is the set of all subsets of A.
We write P(A) for the power set of A,
P(A) = {X : X ⊆ A}
Example: P ({a, b, c}) ={∅,
{a}, {b}, {c},{b, c}, {a, c}, {a, b},{a, b, c}}
True or False: For every set A, A ⊆ P(A)True or False: For every set A, A ∈ P(A)
Theorem
For any finite set A,|P(A)| = 2|A|
There are 193 member nations in the UN. How many ways are there to form acommittee made up of at least two member nations?
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Power Sets
Definition
If A is a set, the power set of A is the set of all subsets of A.
We write P(A) for the power set of A,
P(A) = {X : X ⊆ A}
Example: P ({a, b, c}) ={∅,{a}, {b}, {c},
{b, c}, {a, c}, {a, b},{a, b, c}}
True or False: For every set A, A ⊆ P(A)True or False: For every set A, A ∈ P(A)
Theorem
For any finite set A,|P(A)| = 2|A|
There are 193 member nations in the UN. How many ways are there to form acommittee made up of at least two member nations?
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Power Sets
Definition
If A is a set, the power set of A is the set of all subsets of A.
We write P(A) for the power set of A,
P(A) = {X : X ⊆ A}
Example: P ({a, b, c}) ={∅,{a}, {b}, {c},{b, c}, {a, c}, {a, b},
{a, b, c}}
True or False: For every set A, A ⊆ P(A)True or False: For every set A, A ∈ P(A)
Theorem
For any finite set A,|P(A)| = 2|A|
There are 193 member nations in the UN. How many ways are there to form acommittee made up of at least two member nations?
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Power Sets
Definition
If A is a set, the power set of A is the set of all subsets of A.
We write P(A) for the power set of A,
P(A) = {X : X ⊆ A}
Example: P ({a, b, c}) ={∅,{a}, {b}, {c},{b, c}, {a, c}, {a, b},{a, b, c}}
True or False: For every set A, A ⊆ P(A)True or False: For every set A, A ∈ P(A)
Theorem
For any finite set A,|P(A)| = 2|A|
There are 193 member nations in the UN. How many ways are there to form acommittee made up of at least two member nations?
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Power Sets
Definition
If A is a set, the power set of A is the set of all subsets of A.
We write P(A) for the power set of A,
P(A) = {X : X ⊆ A}
Example: P ({a, b, c}) ={∅,{a}, {b}, {c},{b, c}, {a, c}, {a, b},{a, b, c}}
True or False: For every set A, A ⊆ P(A)True or False: For every set A, A ∈ P(A)
Theorem
For any finite set A,|P(A)| = 2|A|
There are 193 member nations in the UN. How many ways are there to form acommittee made up of at least two member nations?
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Power Sets
Definition
If A is a set, the power set of A is the set of all subsets of A.
We write P(A) for the power set of A,
P(A) = {X : X ⊆ A}
Example: P ({a, b, c}) ={∅,{a}, {b}, {c},{b, c}, {a, c}, {a, b},{a, b, c}}
True or False: For every set A, A ⊆ P(A)True or False: For every set A, A ∈ P(A)
Theorem
For any finite set A,|P(A)| =
2|A|
There are 193 member nations in the UN. How many ways are there to form acommittee made up of at least two member nations?
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Power Sets
Definition
If A is a set, the power set of A is the set of all subsets of A.
We write P(A) for the power set of A,
P(A) = {X : X ⊆ A}
Example: P ({a, b, c}) ={∅,{a}, {b}, {c},{b, c}, {a, c}, {a, b},{a, b, c}}
True or False: For every set A, A ⊆ P(A)True or False: For every set A, A ∈ P(A)
Theorem
For any finite set A,|P(A)| = 2|A|
There are 193 member nations in the UN. How many ways are there to form acommittee made up of at least two member nations?
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Power Sets
Definition
If A is a set, the power set of A is the set of all subsets of A.
We write P(A) for the power set of A,
P(A) = {X : X ⊆ A}
Example: P ({a, b, c}) ={∅,{a}, {b}, {c},{b, c}, {a, c}, {a, b},{a, b, c}}
True or False: For every set A, A ⊆ P(A)True or False: For every set A, A ∈ P(A)
Theorem
For any finite set A,|P(A)| = 2|A|
There are 193 member nations in the UN. How many ways are there to form acommittee made up of at least two member nations?
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Fun Context
Number of subcommittees of the UN consisting of at least two membernations:
2193︸︷︷︸total number of subsets
− 1︸︷︷︸empty set
− 193︸︷︷︸single-element subsets
≈ 2193
population of humans ≈ 7 billion ≈ 232.7
Number of atoms in the earth2 ≈ 1050 ≈ 2166
Number of stars in the observable universe3 ≈ 1021 ≈ 270
Number of atoms in the Milky Way4 ≈ 1068 ≈ 2256
Number of atoms in the known universe5 ≈ 1080 ≈ 2266
How many different subcommittees can Canada be on?
2http://education.jlab.org/qa/mathatom_05.html,http://www.fnal.gov/pub/science/inquiring/questions/atoms.html
3http://scienceline.ucsb.edu/getkey.php?key=37754https://www.quora.com/How-many-atoms-are-there-in-the-milky-way-galaxy5http://www.universetoday.com/36302/atoms-in-the-universe/
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Fun Context
Number of subcommittees of the UN consisting of at least two membernations:
2193︸︷︷︸total number of subsets
− 1︸︷︷︸empty set
− 193︸︷︷︸single-element subsets
≈ 2193
population of humans ≈ 7 billion ≈ 232.7
Number of atoms in the earth2 ≈ 1050 ≈ 2166
Number of stars in the observable universe3 ≈ 1021 ≈ 270
Number of atoms in the Milky Way4 ≈ 1068 ≈ 2256
Number of atoms in the known universe5 ≈ 1080 ≈ 2266
How many different subcommittees can Canada be on?
2http://education.jlab.org/qa/mathatom_05.html,http://www.fnal.gov/pub/science/inquiring/questions/atoms.html
3http://scienceline.ucsb.edu/getkey.php?key=37754https://www.quora.com/How-many-atoms-are-there-in-the-milky-way-galaxy5http://www.universetoday.com/36302/atoms-in-the-universe/
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Fun Context
Number of subcommittees of the UN consisting of at least two membernations:
2193︸︷︷︸total number of subsets
− 1︸︷︷︸empty set
− 193︸︷︷︸single-element subsets
≈ 2193
population of humans ≈ 7 billion ≈ 232.7
Number of atoms in the earth2 ≈ 1050 ≈ 2166
Number of stars in the observable universe3 ≈ 1021 ≈ 270
Number of atoms in the Milky Way4 ≈ 1068 ≈ 2256
Number of atoms in the known universe5 ≈ 1080 ≈ 2266
How many different subcommittees can Canada be on?
2http://education.jlab.org/qa/mathatom_05.html,http://www.fnal.gov/pub/science/inquiring/questions/atoms.html
3http://scienceline.ucsb.edu/getkey.php?key=37754https://www.quora.com/How-many-atoms-are-there-in-the-milky-way-galaxy5http://www.universetoday.com/36302/atoms-in-the-universe/
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Fun Context
Number of subcommittees of the UN consisting of at least two membernations:
2193︸︷︷︸total number of subsets
− 1︸︷︷︸empty set
− 193︸︷︷︸single-element subsets
≈ 2193
population of humans ≈ 7 billion ≈ 232.7
Number of atoms in the earth2 ≈ 1050 ≈ 2166
Number of stars in the observable universe3 ≈ 1021 ≈ 270
Number of atoms in the Milky Way4 ≈ 1068 ≈ 2256
Number of atoms in the known universe5 ≈ 1080 ≈ 2266
How many different subcommittees can Canada be on?
2http://education.jlab.org/qa/mathatom_05.html,http://www.fnal.gov/pub/science/inquiring/questions/atoms.html
3http://scienceline.ucsb.edu/getkey.php?key=37754https://www.quora.com/How-many-atoms-are-there-in-the-milky-way-galaxy5http://www.universetoday.com/36302/atoms-in-the-universe/
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Fun Context
Number of subcommittees of the UN consisting of at least two membernations:
2193︸︷︷︸total number of subsets
− 1︸︷︷︸empty set
− 193︸︷︷︸single-element subsets
≈ 2193
population of humans ≈ 7 billion ≈ 232.7
Number of atoms in the earth2 ≈ 1050 ≈ 2166
Number of stars in the observable universe3 ≈ 1021 ≈ 270
Number of atoms in the Milky Way4 ≈ 1068 ≈ 2256
Number of atoms in the known universe5 ≈ 1080 ≈ 2266
How many different subcommittees can Canada be on?
2http://education.jlab.org/qa/mathatom_05.html,http://www.fnal.gov/pub/science/inquiring/questions/atoms.html
3http://scienceline.ucsb.edu/getkey.php?key=37754https://www.quora.com/How-many-atoms-are-there-in-the-milky-way-galaxy5http://www.universetoday.com/36302/atoms-in-the-universe/
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Fun Context
Number of subcommittees of the UN consisting of at least two membernations:
2193︸︷︷︸total number of subsets
− 1︸︷︷︸empty set
− 193︸︷︷︸single-element subsets
≈ 2193
population of humans ≈ 7 billion ≈ 232.7
Number of atoms in the earth2 ≈ 1050 ≈ 2166
Number of stars in the observable universe3 ≈ 1021 ≈ 270
Number of atoms in the Milky Way4 ≈ 1068 ≈ 2256
Number of atoms in the known universe5 ≈ 1080 ≈ 2266
How many different subcommittees can Canada be on?
2http://education.jlab.org/qa/mathatom_05.html,http://www.fnal.gov/pub/science/inquiring/questions/atoms.html
3http://scienceline.ucsb.edu/getkey.php?key=37754https://www.quora.com/How-many-atoms-are-there-in-the-milky-way-galaxy5http://www.universetoday.com/36302/atoms-in-the-universe/
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
Fun Context
Number of subcommittees of the UN consisting of at least two membernations:
2193︸︷︷︸total number of subsets
− 1︸︷︷︸empty set
− 193︸︷︷︸single-element subsets
≈ 2193
population of humans ≈ 7 billion ≈ 232.7
Number of atoms in the earth2 ≈ 1050 ≈ 2166
Number of stars in the observable universe3 ≈ 1021 ≈ 270
Number of atoms in the Milky Way4 ≈ 1068 ≈ 2256
Number of atoms in the known universe5 ≈ 1080 ≈ 2266
How many different subcommittees can Canada be on?
2http://education.jlab.org/qa/mathatom_05.html,http://www.fnal.gov/pub/science/inquiring/questions/atoms.html
3http://scienceline.ucsb.edu/getkey.php?key=37754https://www.quora.com/How-many-atoms-are-there-in-the-milky-way-galaxy5http://www.universetoday.com/36302/atoms-in-the-universe/
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
True or False: If A is a finite set, then P(A) is finite.
True (even though it can be huuuuuge)
True or False: For every finite set A, |A| ≤ |P(A)|.
True. (Even when A = ∅.)
If A and B are finite sets, calculate the following cardinalities:
|P(A)× B|
= 2|A| · |B|
|P(A× B)|
= 2|A||B|
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
True or False: If A is a finite set, then P(A) is finite.True (even though it can be huuuuuge)
True or False: For every finite set A, |A| ≤ |P(A)|.
True. (Even when A = ∅.)
If A and B are finite sets, calculate the following cardinalities:
|P(A)× B|
= 2|A| · |B|
|P(A× B)|
= 2|A||B|
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
True or False: If A is a finite set, then P(A) is finite.True (even though it can be huuuuuge)
True or False: For every finite set A, |A| ≤ |P(A)|.True. (Even when A = ∅.)
If A and B are finite sets, calculate the following cardinalities:
|P(A)× B|
= 2|A| · |B|
|P(A× B)|
= 2|A||B|
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
True or False: If A is a finite set, then P(A) is finite.True (even though it can be huuuuuge)
True or False: For every finite set A, |A| ≤ |P(A)|.True. (Even when A = ∅.)
If A and B are finite sets, calculate the following cardinalities:
|P(A)× B| = 2|A| · |B|
|P(A× B)|
= 2|A||B|
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
True or False: If A is a finite set, then P(A) is finite.True (even though it can be huuuuuge)
True or False: For every finite set A, |A| ≤ |P(A)|.True. (Even when A = ∅.)
If A and B are finite sets, calculate the following cardinalities:
|P(A)× B| = 2|A| · |B|
|P(A× B)| = 2|A||B|
1. Sets
1.1 Intro-duction toSets
1.2 TheCartesianProduct
1.3 Subsets
1.4 PowerSets
1.5 Union,Intersec-tion,Difference
1.6 Com-plement
1.7 VennDiagrams
1.8 IndexedSets
For Sections 1.5-1.8, see the next set of slides.