Discrrete mathematics for computer science 09sets

Sets

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What is a Set?

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

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

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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”)

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“Determined by its members”

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{7, “Sunday”, π} is a set containing three elements
{7, “Sunday”, π} = {π, 7, “Sunday”, π, 14/2}

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Set Membership

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

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Subset: ⊆

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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

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

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{∅, {1}, {2}, {1,2}}
1 2

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Otherwise infinite
N, Z
{0, 2, 4, 6, 8, …}
(to be continued …}

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Set Constructor

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The set of elements of A of which P is true:

– {x ∈A: P(x)} or {x ∈A | P(x)}

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E.g. the set of even numbers is
{n∈Z: n is even}
= {n∈Z: (∃m∈Z) n = 2m}

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E. g. A×B = {(a,b): a∈A and b∈B}

– Ordered pairs also written

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〈 a,b 〈

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Size of a Finite Set

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

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Size of a Finite Set

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

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Size of a Finite Set

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|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

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|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

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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)

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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)
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