# Operations on Sets II

Intersection of Sets: Given 2 sets A, B, then the set formed by taking the common elements of the set as its element is called intersection of set.

In general, intersection of n-sets is the set in which every element is present in all the n-sets A1 A2 ∩ A3∩ An

$$\bigcup\limits_{i=1}^{n}{{{A}_{i}}}$$  = {x | x Belongs to each set of A1, A2, …, An}

A B ⊆ A, A B ⊆ B, A ∪ B

Two sets A, B are disjoint if A B = ø

N sets A1, A2, A3, … An are pair wise disjoint if Ai Aj = ø, i ≠ j

Note: A B is denoted as AB. Similarly A ∩ B C is denoted as ABC

Properties of intersection of sets:-

1) A A = A (Idempotent property under intersection)

2) A B = B A (Commutative law)

3) A B ∩ C = A (B C) [Associative law]

4) A ø = ø

Note: If 3 sets A, B, C are pair wise disjoint sets then A B C is a null set .But the converse of the statement need not be true.

⇒ In general A1 A2 A3 An = ø

(Converse need not be true)

⇒ Given two non – empty sets A, B and μ assume that A & B are not equal. Then draw Venn diagram in fig cases:

(i) B contains A (A ⊆ B)

A ∪ B = B

A B = A (ii) A contains (B ⊆ A)
A ∪ B = A

A B = B (iii) A, B are disjoint sets

A B = ø (iv) A intersection B non-empty set Note: If A ⊆ B, then A ∪ B = B

A B = A

Result: Union is distributive over intersection

Intersection is distributive over union.

A ∪ B = (B C) = (A ∪ B) ∩ (A ∪ C)

A B = (B ∪ C) = (A B) ∪ (A C)