Union of Sets

Definition of Union
of Sets:

Union of two given sets is the smallest set
which contains all the elements of both the sets.

To find the union of two given sets A and B is a set which consists of all the elements of A and all the elements of B such that no element is repeated.

The symbol for denoting union of sets is ‘∪’.

For example;

Let set A = {2, 4, 5, 6}
and set B = {4, 6, 7, 8}

Taking every element of both the sets A and B, without repeating any element, we get a new set = {2, 4, 5, 6, 7, 8}

This new set contains all the elements of set A and all the elements of set B with no repetition of elements and is named as union of set A and B.

The symbol used for the union of two
sets is ‘∪’.

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

Therefore, A ∪ B = {x : x ∈ A or x ∈ B}

Solved examples to find union of two given sets:

1. If A = {1, 3, 7, 5} and
B = {3, 7, 8, 9}. Find union of two set A and B. 

Solution:

A ∪ B = {1, 3, 5, 7, 8, 9} 

No element is repeated in the union of two sets.
The common elements 3, 7 are taken only once. 

2. Let
X = {a, e, i, o, u} and
Y = {ф}. Find union of two
given sets X and Y. 

Solution:

X ∪ Y = {a, e,
i, o, u} 

Therefore,
union of any set with an empty set is the set itself. 

3.
If set P = {2, 3, 4, 5, 6, 7}, set Q = {0, 3, 6, 9, 12} and set R = {2, 4, 6,
8}.

(i) Find the union of sets P and Q

(ii) Find the union of two set P and R

(iii)
Find the union of the given sets Q and R

Solution:

(i) Union of sets P and Q is P ∪ Q

The smallest set which contains all the
elements of set P and all the elements of set Q is {0, 2, 3, 4, 5, 6, 7, 9, 12}.

(ii) Union of two set P and R is P ∪ R

The smallest set which contains all the
elements of set P and all the elements of set R is {2, 3, 4, 5, 6, 7, 8}.

(iii) Union of the given sets Q and R
is Q ∪ R

The smallest set which contains all the
elements of set Q and all the elements of set R is {0, 2, 3, 4, 6, 8, 9, 12}.

Notes:

A and B are the
subsets of A ∪ B

The union of sets is commutative, i.e., A ∪ B = B ∪ A.

The operations are performed when the sets are
expressed in roster form. 

Some properties of the operation of
union:

(i) A∪B = B∪A                      (Commutative law) 

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

(iii) A ∪ ϕ = A                      (Law of identity element, is the
identity of ∪) 

(iv) A∪A = A
(Idempotent law) 

(v) U∪A = U
(Law of ∪) ∪ is the universal set.

Notes:

A ∪ ϕ = ϕ ∪ A = A i.e. union of any set with the empty set is
always the set itself.