Intersection of sets A & B has all the elements which are common to  set A and set B

It is represented by symbol ∩

 

Let A = {1, 2, 3, 4 } , B = { 3, 4 , 5, 6}

A ∩ B = {3, 4}

Intersection of sets - Venn diagram.jpg

The blue region is A ∩ B


Properties of Intersection

  1. A ∩ B = B ∩ A (Commutative law).
  2. (A  ∩ B) ∩ C = A ∩ (B ∩ C) (Associative law).
  3. ∅ ∩ A = ∅, U ∩ A = A (Law of ∅ and U).
  4. A ∩ A = A (Idempotent law)
  5. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (Distributive law) i. e., ∩ distributes over ∪
    or we can also write it as
    A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

 

Let us discuss these laws

 

Let us take sets

Let A = {1, 2, 3, 4} , B = {3, 4, 5, 6}, C = {6, 7, 8}

and Universal set = U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

 

A ∩ B = B ∩ A (Commutative law).

A B = {1, 2, 3, 4} {3, 4, 5, 6} = {3, 4}

B ∩ A = {3, 4, 5, 6} {1, 2, 3, 4} = {3, 4}

∴ A ∩ B = B ∩ A

 

(A  ∩ B) ∩ C = A ∩ (B ∩ C) (Associative law).

A B = {3, 4}

(A  ∩ B ) ∩ C = {3, 4} ∩ {6, 7, 8} = {} = ∅

 

B ∩ C = {3, 4, 5, 6} ∩ {6, 7, 8} = {6}

A ∩ (B ∩ C) = {1, 2, 3, 4} ∩ {6} = {} = ∅

 

∴ (A  ∩ B) ∩ C = A ∩ (B ∩ C)

 

∩ A = ∅, U ∩ A = A (Law of and U).

In intersection, we have all elements which are common

 

∩ A =

Since ∅ has no elements, there will be no common element between ∅ and A

∴ Intersection of ∅ and A will be ∅

∅ ∩ A = {} ∩ {1, 2, 3, 4} = {}

∴ ∅ ∩ A = ∅

 

U ∩ A = A

Since U has all the elements, the common elements between U and A will be all the elements of set A

∴ Intersection of U and A will be A

U ∩ A = { 1 , 2, 3, 4 , 5, 6, 7, 8, 9, 10} ∩ { 1, 2, 3, 4 }

U ∩ A = {1, 2, 3, 4} = A

∴ U ∩ A = A

 

A ∩ A = A (Idempotent law)

A ∩ A = {1, 2, 3, 4} ∩ {1, 2, 3, 4}

A ∩ A = {1, 2, 3, 4} = A

∴ A ∩ A = A

 

A ∩ (B ∪ C) = ( A ∩ B) ∪ (A ∩ C) (Distributive law) i . e., ∩ distributes over

B ∪ C = {3, 4, 5, 6} ∪ {6, 7, 8} = {3, 4, 5, 6, 7, 8}

A ∩ (B ∪ C) = {1, 2, 3, 4 } ∩ { 3, 4, 5, 6, 7, 8} = {3, 4}

 

A B = {1, 2, 3, 4} {3, 4, 5, 6} = {3, 4}

A C = {1, 2, 3, 4} {6, 7, 8} = {} = ∅

(A ∩ B) ∪ (A ∩ C) = {3, 4} ∪ ∅ = {3, 4}

 

∴ A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

 

Let's also prove

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (Distributive Law)

B ∩ C = {3, 4, 5, 6 } ∩ { 6 , 7, 8} = {6}

A ∪ (B ∩ C) = {1, 2, 3, 4} ∪ {6} = {1, 2, 3, 4, 6}

 

A ∪ B = {1, 2, 3, 4} ∪ {3, 4, 5, 6} = {1, 2, 3, 4, 5, 6}

A ∪ C = {1, 2, 3, 4} ∪ {6, 7, 8} = {1, 2, 3, 4, 6, 7, 8}

(A ∪ B) (A ∪ C) = {1, 2, 3, 4, 5, 6} {1, 2, 3, 4, 6, 7, 8} = {1, 2, 3, 4, 6}

 

∴ A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) 

 

Let us prove distributive law using Venn Diagram

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.