1

Sets

2/10/12

2

What is a Set?

• Informally, a collection of objects, determined by its members, treated as a single mathematical object

• Not a real definition: What’s a collection??

2/10/12

3

Some sets

𝒁 = the set of integers 𝐍 = the set of nonnegative integers

R = the set of real numbers{1, 2, 3}{{1}, {2}, {3}}{Z}∅ = the empty setP({1,2}) = the set of all subsets of {1,2}

= {∅, {1}, {2}, {1,2}}

P( ) = the set of all sets of integers (“the power 𝒁set of the integers”)

2/10/12

4

“Determined by its members”

• {7, “Sunday”, π} is a set containing three elements

• {7, “Sunday”, π} = {π, 7, “Sunday”, π, 14/2}

2/10/12

5

Set Membership

• Let A = {7, “Sunday”, π} • Then 7 ∈A• 8 ∉ A• N ∈ P(Z)

2/10/12

6

Subset: ⊆

• A ⊆ B is read “A is a subset of B” or “A is contained in B”

• (∀x) (x∈A ⇒ x∈B)• N ⊆ Z, {7} ⊆ {7, “Sunday”, π} • ∅ ⊆ A for any set A

(∀x) (x∈∅ ⇒ x∈A)

• A ⊆ A for any set A• To be clear that A ⊆ B but A ≠ B,

write A ⊊ B

• “Proper subset” (I don’t like “⊂”)2/10/12

7

Finite and Infinite Sets

• A set is finite if it can be counted using some initial segment of the integers

• {∅, {1}, {2}, {1,2}} 1 2 3 4• Otherwise infinite• N, Z• {0, 2, 4, 6, 8, …}• (to be continued …}2/10/12

8

Set Constructor

• The set of elements of A of which P is true:– {x ∈A: P(x)} or {x ∈A | P(x)}

• E.g. the set of even numbers is {n∈Z: n is even}

= {n∈Z: (∃m∈Z) n = 2m}

• E. g. A×B = {(a,b): a∈A and b∈B}– Ordered pairs also written 〈 a,b〉

2/10/12

9

Size of a Finite Set

• |A| is the number of elements in A• |{2,4,6}| = ?

2/10/12

10

Size of a Finite Set

• |A| is the number of elements in A• |{2,4,6}| = 3• |{{2,4,6}}| = ?

2/10/12

11

Size of a Finite Set

• |A| is the number of elements in A• |{2,4,6}| = 3• |{{2,4,6}}| = 1• |{N}| = ?

2/10/12

12

Size of a Finite Set

• |A| is the number of elements in A• |{2,4,6}| = 3• |{{2,4,6}}| = 1• |{N}| = 1 (a set containing only one

thing, which happens to be an infinite set)

2/10/12

13

Operators on Sets

• Union: x∈A∪B iff x∈A or x∈B• Intersection: x∈A∩B iff x∈A and x∈B• Complement: x∈B iff x ∉ B• x∈A-B iff x∈A and x∉B• A-B = A\B = A∩B

2/10/12

14

Proof that A ∪ (B∩C) = (A∪B)∩(A∪C)

• x∈A∪(B∩C) iff• x∈A or x∈B∩C (defn of ∪) iff• x∈A or (x∈B and x∈C) (defn of ∩)• Let p := “x∈A”, q := “x∈B”, r := x∈C• Then p ∨ ( q ⋀ r ) ≡

( p ∨ q) ⋀ (p ∨ r) ≡(x∈A or x∈B) and (x∈A or x∈C) iff(x∈A∪B) and (x∈A∪C) iffx∈(A∪B)∩(A∪C)

2/10/12