Introduction to Sets (Optional)

Frank Ma © 2011

A set is a backpack which may or may not contains any items. Introduction to Sets

A set is a backpack which may or may not contains any items. We use capital letters as names of sets.

Introduction to Sets

A set is a backpack which may or may not contains any items. We use capital letters as names of sets. For example, A = { } ≡ Ф (phi) – which is the empty setB = {wallet, my car–key, math–book} C = {my car–key}D = {1, 2, 3, ….} = {all positive integer}

Introduction to Sets

A set is a backpack which may or may not contains any items. We use capital letters as names of sets. For example, A = { } ≡ Ф (phi) – which is the empty setB = {wallet, my car–key, math–book} C = {my car–key}D = {1, 2, 3, ….} = {all positive integer}

Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items.

Introduction to Sets

A set is a backpack which may or may not contains any items. We use capital letters as names of sets. For example, A = { } ≡ Ф (phi) – which is the empty setB = {wallet, my car–key, math–book} C = {my car–key}D = {1, 2, 3, ….} = {all positive integer}

Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items. Each item of a set is called an element of that set.

Introduction to Sets

A set is a backpack which may or may not contains any items. We use capital letters as names of sets. For example, A = { } ≡ Ф (phi) – which is the empty setB = {wallet, my car–key, math–book} C = {my car–key}D = {1, 2, 3, ….} = {all positive integer}

Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items. Each item of a set is called an element of that set. We write x ϵ S if x is an element the set S.

Introduction to Sets

A set is a backpack which may or may not contains any items. We use capital letters as names of sets. For example, A = { } ≡ Ф (phi) – which is the empty setB = {wallet, my car–key, math–book} C = {my car–key}D = {1, 2, 3, ….} = {all positive integer}

Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items. Each item of a set is called an element of that set. We write x ϵ S if x is an element the set S. Hence from the above sets we’ve that

Introduction to Sets

my car–keyϵ B

A set is a backpack which may or may not contains any items. We use capital letters as names of sets. For example, A = { } ≡ Ф (phi) – which is the empty setB = {wallet, my car–key, math–book} C = {my car–key}D = {1, 2, 3, ….} = {all positive integer}

Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items. Each item of a set is called an element of that set. We write x ϵ S if x is an element the set S. Hence from the above sets we’ve that

Introduction to Sets

my car–keyϵ B my car–keyϵ C

A set is a backpack which may or may not contains any items. We use capital letters as names of sets. For example, A = { } ≡ Ф (phi) – which is the empty setB = {wallet, my car–key, math–book} C = {my car–key}D = {1, 2, 3, ….} = {all positive integer}

Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items. Each item of a set is called an element of that set. We write x ϵ S if x is an element the set S. Hence from the above sets we’ve that

Introduction to Sets

my car–keyϵ B my car–keyϵ C my car–keyϵ A

A set is a backpack which may or may not contains any items. We use capital letters as names of sets. For example, A = { } ≡ Ф (phi) – which is the empty setB = {wallet, my car–key, math–book} C = {my car–key}D = {1, 2, 3, ….} = {all positive integer}

Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items. Each item of a set is called an element of that set. We write x ϵ S if x is an element the set S. Hence from the above sets we’ve that

Introduction to Sets

my car–keyϵ B my car–keyϵ C my car–keyϵ A

Note that C is not an element of B because C is just anotherbackpack whose content part (or all) of B’s content.

A set is a backpack which may or may not contains any items. We use capital letters as names of sets. For example, A = { } ≡ Ф (phi) – which is the empty setB = {wallet, my car–key, math–book} C = {my car–key}D = {1, 2, 3, ….} = {all positive integer}

Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items. Each item of a set is called an element of that set. We write x ϵ S if x is an element the set S. Hence from the above sets we’ve that

Introduction to Sets

my car–keyϵ B my car–keyϵ C my car–keyϵ A

Note that C is not an element of B because C is just anotherbackpack whose content part (or all) of B’s content. In generalT is a subset of S if every element of T is also an element of S and we write this as T S.

A set is a backpack which may or may not contains any items. We use capital letters as names of sets. For example, A = { } ≡ Ф (phi) – which is the empty setB = {wallet, my car–key, math–book} C = {my car–key}D = {1, 2, 3, ….} = {all positive integer}

Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items. Each item of a set is called an element of that set. We write x ϵ S if x is an element the set S. Hence from the above sets we’ve that

Introduction to Sets

my car–keyϵ B my car–keyϵ C my car–keyϵ A

Note that C is not an element of B because C is just anotherbackpack whose content part (or all) of B’s content. In generalT is a subset of S if every element of T is also an element of S and we write this as T S. So we’ve C but C B.ϵ B

Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa.

Introduction to Sets

Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.

Introduction to Sets

Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.Note that the order of the elements is not important.

Introduction to Sets

Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.Note that the order of the elements is not important.

Introduction to Sets

Some sets of numbers may be represented graphically.

Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.Note that the order of the elements is not important.

Introduction to Sets

Some sets of numbers may be represented graphically.

Example A. Graph the following sets.a. A = {all the x where –2 < x ≤ 6}

Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.Note that the order of the elements is not important.

Introduction to Sets

Some sets of numbers may be represented graphically.

Example A. Graph the following sets.a. A = {all the x where –2 < x ≤ 6}

x–2 6

Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.Note that the order of the elements is not important.

Introduction to Sets

Some sets of numbers may be represented graphically.

Example A. Graph the following sets.a. A = {all the x where –2 < x ≤ 6}

x–2 6

b. B = {all the x where | x | ≤ 4}

Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.Note that the order of the elements is not important.

Introduction to Sets

Some sets of numbers may be represented graphically.

Example A. Graph the following sets.a. A = {all the x where –2 < x ≤ 6}

x–2 6

b. B = {all the x where | x | ≤ 4}

This is the same as { x where –4 ≤ x ≤ 4}

Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.Note that the order of the elements is not important.

Introduction to Sets

Some sets of numbers may be represented graphically.

Example A. Graph the following sets.a. A = {all the x where –2 < x ≤ 6}

x–2 6

b. B = {all the x where | x | ≤ 4}

x–4 4

This is the same as { x where –4 ≤ x ≤ 4}

Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.Note that the order of the elements is not important.

Introduction to Sets

Some sets of numbers may be represented graphically.

Example A. Graph the following sets.a. A = {all the x where –2 < x ≤ 6}

x–2 6

b. B = {all the x where | x | ≤ 4}

x–4 4

This is the same as { x where –4 ≤ x ≤ 4}

Intersection and Union

Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.Note that the order of the elements is not important.

Introduction to Sets

Some sets of numbers may be represented graphically.

Example A. Graph the following sets.a. A = {all the x where –2 < x ≤ 6}

x–2 6

b. B = {all the x where | x | ≤ 4}

x–4 4

This is the same as { x where –4 ≤ x ≤ 4}

Intersection and UnionLet S and T be two sets, S T, read as S intersects T, is the set of common elements of S and T.

Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.Note that the order of the elements is not important.

Introduction to Sets

Some sets of numbers may be represented graphically.

Example A. Graph the following sets.a. A = {all the x where –2 < x ≤ 6}

x–2 6

b. B = {all the x where | x | ≤ 4}

x–4 4

This is the same as { x where –4 ≤ x ≤ 4}

Intersection and UnionLet S and T be two sets, S T, read as S intersects T, is the set of common elements of S and T. Hence {a, b} {b, c} = {b}.

Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.Note that the order of the elements is not important.

Introduction to Sets

Some sets of numbers may be represented graphically.

Example A. Graph the following sets.a. A = {all the x where –2 < x ≤ 6}

x–2 6

b. B = {all the x where | x | ≤ 4}

x–4 4

This is the same as { x where –4 ≤ x ≤ 4}

Intersection and UnionLet S and T be two sets, S T, read as S intersects T, is the set of common elements of S and T. Hence {a, b} {b, c} = {b}.Note that the intersection is a set, not just the element “b”.

Introduction to SetsLet S and T be two sets, S U T , read as S unions T, is the merge–set of the two sets.

Introduction to SetsLet S and T be two sets, S U T , read as S unions T, is the merge–set of the two sets. For the union operation, duplicated items is only counted once.

Introduction to SetsLet S and T be two sets, S U T , read as S unions T, is the merge–set of the two sets. For the union operation, duplicated items is only counted once. Hence {a, b} U {b, c} = {a, b, c}.

Introduction to Sets

Example B. Find and draw A B and A U B given that

A = {all the x where –2 < x ≤ 6}B = {all the x where | x | ≤ 4}

Let S and T be two sets, S U T , read as S unions T, is the merge–set of the two sets. For the union operation, duplicated items is only counted once. Hence {a, b} U {b, c} = {a, b, c}.

Introduction to Sets

Example B. Find and draw A B and A U B given that

A = {all the x where –2 < x ≤ 6}

x–2 6

B = {all the x where | x | ≤ 4}

x–4 4

Let S and T be two sets, S U T , read as S unions T, is the merge–set of the two sets. For the union operation, duplicated items is only counted once. Hence {a, b} U {b, c} = {a, b, c}.

A

B

We have graphically

Introduction to Sets

Example B. Find and draw A B and A U B given that

A = {all the x where –2 < x ≤ 6}

x–2 6

B = {all the x where | x | ≤ 4}

x–4 4

Let S and T be two sets, S U T , read as S unions T, is the merge–set of the two sets. For the union operation, duplicated items is only counted once. Hence {a, b} U {b, c} = {a, b, c}.

A

B

We have graphically

A B is the common or overlapped segment of A with B.–2 4

A B

Introduction to Sets

Example B. Find and draw A B and A U B given that

A = {all the x where –2 < x ≤ 6}

x–2 6

B = {all the x where | x | ≤ 4}

x–4 4

Let S and T be two sets, S U T , read as S unions T, is the merge–set of the two sets. For the union operation, duplicated items is only counted once. Hence {a, b} U {b, c} = {a, b, c}.

A

B

We have graphically

A B is the common or overlapped segment of A with B. A = { B = {–2 < x ≤ 4}

–2 4

A B

Introduction to Sets

x–2 6

x–4 4

A

B

Given that

Introduction to Sets

x–2 6

x–4 4

A

B

A U B consists both portions of either color

Given that

Introduction to Sets

x–2 6

x–4 4

A

B

A U B consists both portions of either color = {–4 < x ≤ 6}.6–4

A U B

Given that

Introduction to Sets

x–2 6

x–4 4

A

B

A U B consists both portions of either color = {–4 < x ≤ 6}.6–4

A U B

Given that

Let’s extend this to the x&y coordinate system.

Introduction to Sets

x–2 6

x–4 4

A

B

A U B consists both portions of either color = {–4 < x ≤ 6}.6–4

A U B

Given that

Let’s extend this to the x&y coordinate system. The basic 2D regions are half–planes and strips.

Introduction to Sets

x–2 6

x–4 4

A

B

A U B consists both portions of either color = {–4 < x ≤ 6}.6–4

A U B

Given that

Let’s extend this to the x&y coordinate system. The basic 2D regions are half–planes and strips.

Example. C. Shade the following sets in the rectangular system. a. A = {(x, y) where x > 0}

Introduction to Sets

x–2 6

x–4 4

A

B

A U B consists both portions of either color = {–4 < x ≤ 6}.6–4

A U B

Given that

Let’s extend this to the x&y coordinate system. The basic 2D regions are half–planes and strips.

Example. C. Shade the following sets in the rectangular system. a. A = {(x, y) where x > 0}Note that there is no mention of y means that y may take on any value.

Introduction to Sets

x–2 6

x–4 4

A

B

A U B consists both portions of either color = {–4 < x ≤ 6}.6–4

A U B

Given that

Let’s extend this to the x&y coordinate system. The basic 2D regions are half–planes and strips.

{(x, y) where x > 0}

Example. C. Shade the following sets in the rectangular system. a. A = {(x, y) where x > 0}Note that there is no mention of y means that y may take on any value.

Introduction to Sets

x–2 6

x–4 4

A

B

A U B consists both portions of either color = {–4 < x ≤ 6}.6–4

A U B

Given that

Let’s extend this to the x&y coordinate system. The basic 2D regions are half–planes and strips.

{(x, y) where x > 0}

Example. C. Shade the following sets in the rectangular system. a. A = {(x, y) where x > 0}Note that there is no mention of y means that y may take on any value. The open region is called a half–plane and the dash–line means exclusion.

Introduction to Setsb. B = {(x, y) where 0 < y ≤ 4}

Introduction to Setsb. B = {(x, y) where 0 < y ≤ 4}

There is no restriction on x so x may take on any value.

Introduction to Setsb. B = {(x, y) where 0 < y ≤ 4}

There is no restriction on x so x may take on any value.

{(x, y) where 0 < y ≤ 4}

Introduction to Setsb. B = {(x, y) where 0 < y ≤ 4}

There is no restriction on x so x may take on any value.

{(x, y) where 0 < y ≤ 4}

The open region we obtained is called a strip.

Introduction to Setsb. B = {(x, y) where 0 < y ≤ 4}

There is no restriction on x so x may take on any value.

{(x, y) where 0 < y ≤ 4}

The open region we obtained is called a strip. Note that the solid represent inclusion.

Introduction to Setsb. B = {(x, y) where 0 < y ≤ 4}

There is no restriction on x so x may take on any value.

{(x, y) where 0 < y ≤ 4}

The open region we obtained is called a strip. Note that the solid represent inclusion.

c. Let C = {(x, y) where –3 < x < 3 and 0 < y ≤ 4} draw B C.

Introduction to Setsb. B = {(x, y) where 0 < y ≤ 4}

There is no restriction on x so x may take on any value.

{(x, y) where 0 < y ≤ 4}

The open region we obtained is called a strip. Note that the solid represent inclusion.

c. Let C = {(x, y) where –3 < x < 3 and 0 < y ≤ 4} draw B C. This is the overlap of two strips.

Introduction to Setsb. B = {(x, y) where 0 < y ≤ 4}

There is no restriction on x so x may take on any value.

{(x, y) where 0 < y ≤ 4}

The open region we obtained is called a strip. Note that the solid represent inclusion.

c. Let C = {(x, y) where –3 < x < 3 and 0 < y ≤ 4} draw B C. This is the overlap of two strips.B is the same as the above.

B

Introduction to Setsb. B = {(x, y) where 0 < y ≤ 4}

There is no restriction on x so x may take on any value.

{(x, y) where 0 < y ≤ 4}

The open region we obtained is called a strip. Note that the solid represent inclusion.

c. Let C = {(x, y) where –3 < x < 3 and 0 < y ≤ 4} draw B C. This is the overlap of two strips.B is the same as the above.

BC is the vertical strip where–3 < x < 3.

C

Introduction to Setsb. B = {(x, y) where 0 < y ≤ 4}

There is no restriction on x so x may take on any value.

{(x, y) where 0 < y ≤ 4}

The open region we obtained is called a strip. Note that the solid represent inclusion.

c. Let C = {(x, y) where –3 < x < 3 and 0 < y ≤ 4} draw B C. This is the overlap of two strips.B is the same as the above.

BC is the vertical strip where–3 < x < 3. The overlap is the rectangular region. B C

C

Introduction to Setsb. B = {(x, y) where 0 < y ≤ 4}

There is no restriction on x so x may take on any value.

{(x, y) where 0 < y ≤ 4}

The open region we obtained is called a strip. Note that the solid represent inclusion.

c. Let C = {(x, y) where –3 < x < 3 and 0 < y ≤ 4} draw B C. This is the overlap of two strips.B is the same as the above.C is the vertical strip where–3 < x < 3. The overlap is the rectangular region. Note that all the cornersare excluded.

B

B C

C

Introduction to Setsb. B = {(x, y) where 0 < y ≤ 4}

There is no restriction on x so x may take on any value.

{(x, y) where 0 < y ≤ 4}

The open region we obtained is called a strip. Note that the solid represent inclusion.

c. Let C = {(x, y) where –3 < x < 3 and 0 < y ≤ 4} draw B C. This is the overlap of two strips.B is the same as the above.C is the vertical strip where–3 < x < 3. The overlap is the rectangular region. Note that all the cornersare excluded. You Do: Label the corners.

B

C

B C