Sets
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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??
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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”)
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“Determined by its members”
• {7, “Sunday”, π} is a set containing three
elements
• {7, “Sunday”, π} = {π, 7, “Sunday”, π,
14/2}
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Set Membership
•
•
•
•
Let A = {7, “Sunday”, π}
Then 7 ∈A
8∉A
N ∈ P(Z)
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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 “⊂”)
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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 …}
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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〉
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Size of a Finite Set
• |A| is the number of elements in A
• |{2,4,6}| = ?
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Size of a Finite Set
• |A| is the number of elements in A
• |{2,4,6}| = 3
• |{{2,4,6}}| = ?
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Size of a Finite Set
•
•
•
•
|A| is the number of elements in A
|{2,4,6}| = 3
|{{2,4,6}}| = 1
|{N}| = ?
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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)
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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
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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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