Discrrete mathematics for computer science 09sets

Sets 2/10/12 What is a Set? • Informally, a collection of objects, determined by its members, treated as a single mathematical object • 2/10/12 Not a real definition: What’s a collection?? 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”) 2/10/12 “Determined by its members” • • 2/10/12 {7, “Sunday”, π} is a set containing three elements {7, “Sunday”, π} = {π, 7, “Sunday”, π, 14/2} Set Membership • • • • 2/10/12 Let A = {7, “Sunday”, π} Then ∈A 8∉A N ∈ P(Z) Subset: ⊆ • • • • • • • 2/10/12 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 “⊂”) Finite and Infinite Sets • A set is finite if it can be counted using some initial segment of the integers • • • • • {∅, {1}, {2}, {1,2}} 2/10/12 Otherwise infinite N, Z {0, 2, 4, 6, 8, …} (to be continued …} 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} = {nZ: (mZ) n = 2m} E g AìB = {(a,b): a∈A and b∈B} – Ordered pairs also written 2/10/12 〈 a,b 〈 Size of a Finite Set • • 2/10/12 |A| is the number of elements in A |{2,4,6}| = ? Size of a Finite Set • • • 2/10/12 |A| is the number of elements in A |{2,4,6}| = |{{2,4,6}}| = ? 10 Size of a Finite Set • • • • 2/10/12 |A| is the number of elements in A |{2,4,6}| = |{{2,4,6}}| = |{N}| = ? 11 Size of a Finite Set • • • • |A| is the number of elements in A |{2,4,6}| = |{{2,4,6}}| = |{N}| = (a set containing only one thing, which happens to be an infinite set) 2/10/12 12 Operators on Sets • • • • • 2/10/12 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 13 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 (p ⋀ ∨ r) q) ≡ ⋀ (x∈A or x∈B) and (x∈A or x∈C) (x∈A∪B) and (x∈A∪C) (p ∨ r) ≡ iff iff x∈(A∪B)∩(A∪C) 2/10/12 14 ... 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...What is a Set? • Informally, a collection of objects, determined by its members, treated as a single mathematical object