Discrrete mathematics for computer science 09sets

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

� = 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”)

“Determined by its members”

• {7, “Sunday”, π} is a set containing three elements

• {7, “Sunday”, π} = {π, 7, “Sunday”, π, 14/2}

Set Membership

• Let A = {7, “Sunday”, π}

• Then 7 ∈A

• 8 ∉ A

• N ∈ P(Z)

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 “⊂”)

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

• {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}

= {n∈Z: (∃m∈Z) n = 2m}

• E g A×B = {(a,b): a∈A and b∈B}

– Ordered pairs also written 〈 a,b 〈

Size of a Finite Set

• |A| is the number of elements in A

• |{2,4,6}| = ?

Size of a Finite Set

• |A| is the number of elements in A

• |{2,4,6}| = 3

• |{{2,4,6}}| = ?

Size of a Finite Set

• |A| is the number of elements in A

• |{2,4,6}| = 3

• |{{2,4,6}}| = 1

• |{N }| = ?

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)

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

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