Sets

Definition

A set is a well-defined collection of objects. Each object of a set is called an element.

Methods of Representing a Set

• Roster or Tabular Form: List elements within braces { }.
• Set-builder Form: {x : P(x) holds}, meaning "the set of all x such that property P(x) holds".

Types of Sets

• Empty Set: No elements.
• Singleton Set: One element.
• Finite Set: Fixed number of elements.
• Infinite Set: Not finite.
• Equal Sets: A = B if every element of A is in B and vice versa.

Subsets

A ⊆ B if every element of A is also in B.

• Every set is a subset of itself.
• The empty set ∅ is a subset of every set.

Intervals as Subsets of R

• Closed: [a, b] = {x ∈ R : a ≤ x ≤ b}
• Open: (a, b) = {x ∈ R : a < x < b}
• Semi-open: (a, b] = {x ∈ R : a < x ≤ b}, [a, b) = {x ∈ R : a ≤ x < b}

Power Set

P(A) = collection of all subsets of A. If A has n elements, then P(A) has 2ⁿ elements.

Universal Set

Includes all elements under consideration. Example: If A = {1,2,3}, B = {3,4,7}, C = {2,8,9}, then U = {1,2,3,4,7,8,9}.

Venn Diagrams

Relationships between sets can be represented visually using Venn diagrams.

Operations on Sets

• Union: A ∪ B = elements in A or B.
• Intersection: A ∩ B = elements common to A and B.
• Difference: A – B = elements in A not in B.

Properties of Union

• A ∪ B = B ∪ A (Commutative)
• (A ∪ B) ∪ C = A ∪ (B ∪ C) (Associative)
• A ∪ ∅ = A (Identity)
• A ∪ A = A (Idempotent)
• U ∪ A = U (Universal)

Properties of Intersection

• A ∩ B = B ∩ A (Commutative)
• (A ∩ B) ∩ C = A ∩ (B ∩ C) (Associative)
• ∅ ∩ A = ∅, U ∩ A = A
• A ∩ A = A (Idempotent)
• A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (Distributive)

Important Results

• If A ∩ B = ∅, then n(A ∪ B) = n(A) + n(B)
• If A ∪ B ≠ ∅, then n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
• For A, B, C finite: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C)

Complement of a Set

If U is the universal set and A ⊆ U, then A' = U – A = {x ∈ U: x ∉ A}.

Properties of Complements

• Complement Laws: A ∪ A' = U, A ∩ A' = ∅
• De Morgan’s Laws: (A ∪ B)' = A' ∩ B', (A ∩ B)' = A' ∪ B'
• Double Complement: (A')' = A
• Laws of ∅ and U: ∅' = U, U' = ∅