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

Properties of Complement on Sets

• 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'

Venn-Diagram Method

the shaded region is (AUB)'

the shaded region is B'

the shaded region is B'

Thus
(AUB)'=A'∩B'