Intersection of Sets

The intersection of two sets A and B ( denoted by A∩B ) is the set of all elements that is common to both A and B. In mathematical form,

For two sets A and B,
    A∩B = { x: x∈A and x∈B }

Shaded part demonstrates the intersection between two sets

Similarly for three sets A, B and C,

A∩B∩C = { x: x∈A and x∈B and x∈C }

Venn diagram showing intersection of three sets where shaded part denotes the intersection

Intersection of Sets Examples

Example #1: Intersection of Two Sets With Venn Diagram

If A = {a, b, c, d, e} and B = {d, e, f, g}, find A∩B.

Here,
    A = {a, b, c, d, e}
    B = {d, e, f, g}
Now,
  A∩B = {a, b, c, d, e} ∩ {d, e, f, g}
∴ A∩B = {d, e}

Example showing of intersection of two sets where shaded part shows the intersection.

Example #2

Suppose A = { x: x is an integer between 1 and 7} and B = { x: x is an integer between 4 and 9} then, find A∩B.

Here,
   A = {2, 3, 4, 5, 6}
   B = {5, 6, 7, 8}
 ∴ A∩B = {5, 6} = { x: x is an integer between 4 and 7}

Example #3: Intersection of Disjoint Sets

If A = {a, b, c} and B = {d, e, f, g}, find A∩B.

Here,
   A∪B = {a, b, c} ∩ {d, e, f, g}
∴ A∪B = {} = ϕ

Intersection of disjoint sets is an empty set

Example #4

If A = { x: x is an integer} and B = { x: x is an even integer} then, find A∩B.

Here,
  A = {...,  -3, -2, -1, 0, 1, 2, ...}
  B = {...,  -2, 0, 2, ...}
∴ A∩B = {..., -4, -2, 0, 2, ... } = {x: x is an even integer}

Example #5: Intersection of Three Sets With Venn Diagram

If A = {a, b, c, d, e}, B = {d, e, f, g} and C = {c, e, f, h, i}, find A∩B∩C.

Here,
  A∩B∩C = {a, b, c, d, e} ∩ {d, e, f, g} ∩ {c, e, f, h, i}

The common element among all three sets is 'e'
  ∴ A∩B = {e}

Example showing intersection of three set with venn diagram.

Properties of Intersection of Sets

1. Commutative Property

If A and B are two sets then, A∩B = B∩A

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

 
    A∩B = {3, 4}  
    B∩A = {4, 3}

∴ A∩B = B∩A

2. Associative Property

If A, B and C are three sets then, A∩(B∩C)= (A∩B)∩C.

Suppose,
   A = {1, 2, 3, 4}
   B = {3, 4, 5, 6}
   C = {3, 4, 1, 5, 8, 9} 

   B∩C = {3, 4, 5} 
   A∩B = {3, 4}
 
   A∩(B∩C) = {3, 4} 
   (A∩B)∩C = {3, 4}

∴ A∩(B∩C) = (A∩B)∩C

3. Identity Property

The intersection of a set and the empty set is always the empty set, i.e, A∩ϕ = ϕ.

Suppose,
  A = {a, b, c}
  B = ϕ = { } 

  A∩B = {1, 2, 3, } ∩ {} = {} = ϕ   
∴ A∪ϕ = ϕ