# Algebra of Sets

#### Properties of Union on Sets

• A U B = B U A
• A U A= A
• A U Φ = A
• A U U = U
• If A ⊂ B then A U B = B
• ( A U B ) U C = A U (B U C)

#### Properties of Intersection on Sets

• A ∩ B= B ∩ A
• A ∩ A= A
• A ∩ Φ = Φ
• A ∩ U = A
• (A ∩ B) ∩ C= A ∩ (B ∩ C)
• If A ⊂ B then A ∩ B = A

• A U A' = U
• A ∩ A' = Φ
• U' = Φ
• Φ ' =U

#### Algebra of sets

The algebra of sets develops and describes the basic properties and laws on sets. The common law on set-theoretical operations of union, intersection, complementation and other relations are mentioned below.

#### Properties on Set Equality and Set Inclusion

Let A, B and C be subsets of a universal set U then

• A = B ⇒ B = A
• A = B, B = C ⇒ A = C
• A ⊂ B , B ⊂ C ⇒ A ⊂ C
• A ⊂ Φ ⇒ A = Φ

#### Laws of algebra on sets

Let A, B and C be subsets of a universal set U then

 Laws Identities over union Identities over Intersection Idempotent Laws A U A = A A ∩ A = A Identities Laws A U U = U and A U Φ = A A ∩ U = A and A ∩ Φ = Φ Complement Laws AU A' = U and (A')'=A A ∩ A' = Φ and U'=Φ Commutative Laws A U B = B U A A ∩ B = B ∩ A Associative Laws ( A U B )U C = AU (BU A) (A∩ B)∩ C=A∩ (B∩ A) Distributive Laws AU (B∩ C)=( A U B )∩ (AU C) A∩ (BU C)=(A∩ B)U (A∩ C) De-Morgan’s Laws (AU B)'=A' ∩ B' (AU B)'=A' ∩ B'

#### Theorem 1

Prove that (AUB)'=A'∩B'

#### Set Builder Method

We start by LHS, then

 (AUB)' =U-(AUB) ={x:x∈U and x∉(AUB)} ={x:x∈U and (x∉ A and x∉B)} ={x: (x∈U and x∉ A) and (x∈U and x∉B)} ={x:x∈A' and x∈B'} =A'∩B'

Thus
(AUB)'=A'∩B'

#### Membership Tabular Method

 A B AUB (AUB)' A' B' A'∩B' 1 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 1

Thus
(AUB)'=A'∩B'