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Sets

Sets. 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??. Some sets.

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Sets

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  1. Sets

  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??

  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 set P({1,2}) = the set of all subsets of {1,2} = {∅, {1}, {2}, {1,2}} P(𝒁) = the set of all sets of integers (“the power set of the integers”)

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

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

  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 “⊂”)

  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 …}

  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〉

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

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

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

  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)

  13. Operators on Sets Union: x∈A∪Biffx∈A or x∈B Intersection: x∈A∩Biffx∈A and x∈B Complement: x∈Biffx∉ B x∈A-B iffx∈A and x∉B A-B = A\B = A∩B

  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) iff x∈(A∪B)∩(A∪C)

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