Operations on Sets

When we perform the operation of addition , multiplication on the pair of numbers , we will get another number. Similarly when we perform the operations on two sets , we will get another set. We will now define certain operations on sets and examine their properties.

Henceforth, we will refer all our sets as subsets of some universal set.

Union of sets

Let $A$ and $B$ be any two sets. Union of sets means taking all the elements of $A$ and $B$ keeping the common elements only once.$\cup$ is the symbol to denote the union. Symbolically, we write $A\cup B$ , it is read as "$A$ union $B$ ".

Example : 1

Let $A=\{2,4,6,8\}$ and $B=\{6,8,10,12\}$

$A\cup B=\{2,4,6,8,10,12\}$ . Here the common elements of these two sets are $6,8$ which are taken only once while showing $A\cup B$..

Example : 2

Let $A=\{1,3,5,7,9,11,13\}$ and $B=\{1,5,9,13\}$

Here $B$ is a subset of $A$. Union of sets $A$ and its subset $B$ is the set $A$ itself.

$A\cup B=\{1,3,5,7,9,11,13\}$

i.e if $B\subset A$ then $A\cup B=A$

Definition

The union of two sets A and B is the set C which consists of all those elements which are either in A or in B (including

those which are in both). In symbols, we write $A\cup B=\{x:x\}$

A

∪

B = {

x

:

x

∈

A or

x

∈

B }

The union of two sets can be represented by

a Venn diagram as shown in Fig 1.4.

The shaded portion in Fig 1.4 represents

A

∪

B