Union Of Sets And Venn Diagram Examples

Union of sets:

The basic operations that can be performed on sets are as follows:

\(~~~~~~~~~~~\)i) Union of sets

\(~~~~~~~~~~~\)ii) Intersection of sets

\(~~~~~~~~~~~\)iiii) Difference of sets

In mathematics, we perform certain operations like addition, subtraction, multiplication, etc. These operators generally take two or more operands and give a result based on the operation performed. Similarly, in set theory usually certain operations are performed on two or more sets to get a new set of elements based on the operation performed.

In the upcoming discussions we will study about the union thoroughly. Let us consider a universal set U such that A and B are the subsets of this universal set. The union of two sets A and B is defined as the set of all the elements which lie in set A and set B or both the elements in A and B altogether. The union of set is denoted by the symbol ‘∪’.

union of sets

In the given Venn diagram, the red colored portion represents the union of both the sets A and B.

Thus, the union of two sets A and B is given by a set C, which is also a subset of the universal set U such that C consists of all those elements or members which are either in set A or set B or in both A and B i.e.,

A∪B={x:x ∈A or x∈B}

Let us go through an example to make it clearer.

Example: Let U be a universal set consisting of all the natural numbers less than 20 and set A and B be a subset of U defined as A={2,5,9,15,19} and B={8,9,10,13,15 ,17}. Find A∪B.

Solution: Given U={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}

A={2,5,9,15,19}

B={8,9,10,13,15 ,17}

A∪B={2,5,8,9,10,13,15,17,19}

This can be represented using the following Venn diagram:

union of sets

Properties of Union of two Sets:

i)   Commutative Law: The union of two or more sets follows the commutative law i.e., if we have two sets A and B then,

A∪B=B∪A

ii) Associative Law: The union operation follows the associative law i.e., if we have three sets A, B and C then

(A∪B)∪C = A∪(B∪C)

iii)  Identity Law: The union of an empty set with any set A gives the set itself i.e.,

A∪∅=A

iv)  Idempotent Law: The union of any set A with itself gives the set A i.e.,

A∪A=A

v)  Domination Law: The union of a universal set U with its subset A gives the universal set itself.

 A∪U=U

This is all about the union operation. Quench your thirst for knowledge with us at BYJU’s. To learn about intersection of sets, download BYJU’S-the learning app.


Practise This Question

A die is rolled and the outcomes are observed. Event A is an even number turns up and event B is a prime number turns up. Match the events on left with the outcomes on right.

1.Event A or BP.14, 6}2.Event A and BQ.12}3.Event A but not BR.12,3,4,5,6,}4.Complementary event of BS.11,4,6}