Sets 2/10/121

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?? 2/10/122

Some sets = the set of integers = the set of nonnegative integers R = the set of real numbers {1, 2, 3} {{1}, {2}, {3}} {Z} ∅ = the empty set P({1,2}) = the set of all subsets of {1,2} = { ∅, {1}, {2}, {1,2}} P() = the set of all sets of integers (“the power set of the integers”) 2/10/123

“Determined by its members” {7, “Sunday”, π} is a set containing three elements {7, “Sunday”, π} = {π, 7, “Sunday”, π, 14/2} 2/10/124

Set Membership Let A = {7, “Sunday”, π} Then 7 ∈ A 8 ∉ A N ∈ P(Z) 2/10/125

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 “ ⊂ ”) 2/10/126

Finite and Infinite Sets A set is finite if it can be counted using some initial segment of the integers { ∅, {1}, {2}, {1,2}} Otherwise infinite N, Z {0, 2, 4, 6, 8, …} (to be continued …} 2/10/127

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 〉 2/10/128

Size of a Finite Set |A| is the number of elements in A |{2,4,6}| = ? 2/10/129

Size of a Finite Set |A| is the number of elements in A |{2,4,6}| = 3 |{{2,4,6}}| = ? 2/10/1210

Size of a Finite Set |A| is the number of elements in A |{2,4,6}| = 3 |{{2,4,6}}| = 1 |{N}| = ? 2/10/1211

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) 2/10/1212

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 2/10/1213

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)iff x ∈ (A ∪ B)∩(A ∪ C) 2/10/1214