Intersection of Sets

The intersection of two or more sets is a set that contains all the elements common to the original sets.

Symbol

In set theory, the intersection of sets is denoted by the symbol ‘∩.’ The intersection of ‘n’ number of sets is expressed as Set 1 ∩ Set 2 ∩ Set 3 ∩ … ∩ Set n.

For Two Sets

Formula

The formula for the intersection of two sets (A intersection B) is given by

A ∩ B = {x | x ∈ A and x ∈ B}

This means x is an element of A ∩ B if and only if x is an element of A and x is an element of B.

If A and B are two sets, then the intersection of A and B is written as A ∩ B and read as ‘A intersection B.’

Example

If A = {3, 7} and B = {4, 7, 9} are two sets, then A ∩ B = {7}, which is the only element common to sets A and B.

Number of Elements in Set A ∩ B

If A, B, and C are three sets, then the formula to find the cardinality of the intersection of two sets is given by:

n(A ∩ B) = n(A) + n(B) – n(A ∪ B)

Here,

n(A ∪ B) = Cardinality of set  A ∪ B

n(A) = Cardinality of set A

n(B) = Cardinality of set B

n(A ∩ B) = Cardinality of set A ∩ B

For Three Sets

If A, B, and C are three sets, their intersection is written as A ∩ B ∩ C. It is the set whose all elements are common to A, B, and C.

Formula

The formula for the intersection of three sets (A intersection B intersection C) is given by

A ∩ B ∩ C = {x | x ∈ A and x ∈ B and x ∈ C}

Examples

If A = {3, 7, 9}, B = {7, 9, 11, 13}, and C = {4, 7, 9} are three sets, then A ∩ B ∩ C is the element common to the sets A, B, and C. 

Here, A ∩ B ∩ C = {7, 9}, which are the two elements common to sets A, B, and C

Intersection of Sets and Venn Diagram

In the Venn diagram, the shaded region illustrates the intersection of sets A and B. It covers all the elements common to both sets.

Similarly, a Venn diagram can be drawn for the intersection of three sets: A, B, and C.

Complement of Intersection of Sets

The complement of the intersection of sets is a set containing all the elements in the universal set but absent in A ∩ B. It is represented by (A ∩ B)’ and is read as ‘A intersection B complement.’ 

In the Venn diagrams below, the shaded regions mark the complement of the intersection of three sets: A, B, and C.

For example,

If A = {8, 16, 25}, B = {10, 12, 16}, and U = {6, 8, 10, 12, 16, 20, 25}, then A ∩ B = {16}

Then, (A ∩ B)’ = U – (A ∩ B) = {6, 8, 10, 12, 16, 20, 25} – {16} = {6, 8, 10, 12, 20, 25}

Properties

Commutative 

If A and B are two sets, then according to the commutative law 

A ∩ B = B ∩ A

Associative 

If A, B, and C are three sets, then according to the associative law  

(A ∩ B) ∩ C = A ∩ (B ∩ C)

Distributive 

If A, B, and C are three sets, then according to the distributive law for the intersection of sets

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Empty Set (ɸ)

The intersection of any set with an empty set produces the empty set.

A ∩ ɸ = ɸ 

If A  = {2,10, 25} and ɸ is an empty set, then A ∩ ɸ = {2, 10, 25} ∩ { } = { } = ɸ

Universal Set (U)

The intersection of any set with the universal set produces the original set.

A ∩ U = U

If A  = {2,10, 25} and U = {1, 2, 5, 10, 15, 20, 25}, then A ∩ U = {2,10, 25} ∩ {1, 2, 5, 10, 15, 20, 25} = {1, 2, 5, 10, 15, 20, 25} = U

Idempotent 

It states that the intersection of a set with itself results in the same original set.

A ∩ A = A

Union and Intersection of Sets

Union of Sets	Intersection of Sets
The union of sets is a new set containing all the given set elements.	The intersection of sets is a new set that contains all common elements of the given sets.
Denoted by the symbol ‘∪.’	Denoted by the symbol ‘∩.’
Written as A ∪ B and read as ‘A union B.’	Written as A ∩ B and read as ‘A intersection B.’
In set builder notation, it can be expressed asA ∪ B = {x | x ∈ A or x ∈ B}	In set builder notation, it can be expressed asA ∩ B = {x | x ∈ A and x ∈ B}
Example:If A = {3, 13} and B = {3, 7, 17}, then A ∪ B = {3, 7, 13, 17}	Example:If A = {3, 13} and B = {3, 7, 17}, then A ∩ B = {3}