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

Fig 1 - Union of sets

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 $B\subset A$ it is read as $B$ is a subset of $A$.

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 $A\cup B=\{x:x\in A\ or\ x\in B\}$. The union of two sets can be represented by a Venn diagram as shown in Fig 1

The shaded portion in Fig 1 represents $A\cup B$.

Intersection of sets

Fig 2 - Intersection of sets

If two sets $A$ and $B$ are given, then the intersection of $A$ and $B$ is the set of all elements which are common to both $A$ and $B$. Intersection is denoted by the symbol $\cap$. Symbolically, we write $A\cap B=\{x:x\in A\ and\ x\in B\}$ where $x$ is the common element of both sets $A$ and $B$.

The intersection of two sets can be represented by a Venn diagram as shown in Fig 2.

Example:

Let $A=\{1,2,3,4,5,6,7,8,9,10,11,12\}$ and $B=\{6,8,10,12\}$

Here $A\cap B=\{6,8,10,12\}$ also $B\subset A$ and $A\cap B=B$.

Fig 3 - Disjoint set

A pair of sets which does not have any common element are called disjoint sets.

For example, set $A=\{2,3,4,5\}$ and $B=\{6,8,10,12\}$ are disjoint sets as there are no

elements common between $A$ and $B$. $A\cap B=\emptyset$ meaning empty set.

This is represented by a Venn diagram as shown in Fig 3.

Difference of sets

Fig 4 - Difference of sets

If there are two sets $A$ and $B$ , then the difference of two sets $A$ and $B$ is equal to the set which consists of elements present in $A$ but not in $B$ . It is represented by $A-B$ read as $A$ minus $B$.

Example: If $A=\{2,3,4,5,6,8\}$ and $B=\{6,8,10,12\}$ are two sets.

Then, the difference of set $A$ and set $B$ is given by $A-B=\{2,3,4,5\}$

We can rewrite the definition of difference as $A-B=\{x:x\in A\ and\ x\notin B\}$

The difference of two sets $A$ and $B$ can be represented by Venn diagram as shown in Fig 4. The shaded portion represents the difference of the two sets $A$ and $B$.

Fig 5 - Difference of sets

Then, the difference of set $B$ and set $A$ is given by $B-A=\{10,12\}$

We can rewrite the definition of difference as $B-A=\{x:x\in B\ and\ x\notin A\}$

The difference of two sets $B$ and $A$ can be represented by Venn diagram as

shown in Fig 5. The shaded portion represents the difference of the two sets $B$ and $A$.