Intersection of Sets

Definition of Intersection of Sets:

Intersection of two given sets is the
largest set which contains all the elements that are common to both the sets.

To find the intersection of two given sets A and B is a set which consists of all the elements which are common to both A and B.

The symbol for denoting intersection of sets is ‘∩‘.

For example:

Let set A = {2, 3, 4, 5, 6}

and set B = {3, 5, 7, 9}

In this two sets, the elements 3 and 5 are common. The set containing these common elements i.e., {3, 5} is the intersection of set A and B.

The symbol used for the intersection of two sets is ‘∩‘.

Therefore, symbolically, we write intersection of the two sets A and B is A ∩ B which means A intersection B.

The intersection of two sets A and B is represented as A ∩ B = {x : x ∈ A and x ∈ B}

Solved examples to find intersection of two given sets:

1. If A = {2, 4, 6, 8, 10} and B = {1, 3, 8, 4, 6}. Find intersection of two set A and B. 

Solution:

A ∩ B = {4, 6, 8}

Therefore, 4, 6 and 8 are the common
elements in both the sets.

2. If X = {a, b, c} and Y = {ф}. Find intersection of two given sets X and Y. 

Solution:

X ∩ Y = { }

3. If set A = {4, 6, 8, 10, 12},
set B = {3, 6, 9, 12, 15, 18} and set C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

(i) Find
the intersection of sets A and B.

(ii) Find
the intersection of two set B and C.

(iii)
Find the intersection of the given sets A and C.

Solution:

(i) Intersection of sets A and B is A ∩ B

Set of all the elements which are
common to both set A and set B is {6, 12}.

(ii) Intersection of two set B and C is B ∩ C

Set of all the elements which are
common to both set B and set C is {3, 6, 9}.

(iii) Intersection of the given sets A and C is A ∩ C

Set of all the elements which are
common to both set A and set C is {4, 6, 8, 10}.

Notes:

A ∩ B is a subset of A
and B.

Intersection of a set is commutative, i.e., A ∩ B = B ∩ A.

Operations are performed when the set is
expressed in the roster form.

Some properties of the operation of
intersection

(i) A∩B = B∩A (Commutative law)

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

(iii) ϕ ∩ A = ϕ (Law of ϕ)

(iv) U∩A = A (Law of ∪)

(v) A∩A = A (Idempotent law)

(vi) A∩(B∪C) = (A∩B) ∪ (A∩C) (Distributive law) Here ∩ distributes over ∪

Also, A∪(B∩C) = (AUB) ∩ (AUC) (Distributive law) Here ∪ distributes over ∩

Notes:

A ∩ ϕ = ϕ ∩ A = ϕ i.e. intersection of
any set with the empty set is always the empty set.