MyRank

Click here to go to MyRank

Monday, October 17, 2016

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 ∩ A∩ A3 … ∩ An

= {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:-

∩ 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 3sets 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)

∪ B = B

∩ B = A
(II) A contains (B  A) 

∪ B = A

∩ B = B
(III) A, B are disjoint sets

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

∩ B = A

Result: - Union is distributive over intersection

Intersection is distributive over union.

∪ B = (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

∩ B = (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

No comments:

Post a Comment