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 … ∩ A_n

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

A ∩ B ⊆ A, A ∩ B ⊆ B, A ∪ B

Two sets A, B are disjoint if A ∩ B = ø

N sets A₁, A₂, A₃, … A_n 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 … ∩ A_n = 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 setNote: 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)