# Union of Sets

## Definition of Union of Sets

The Union of two Sets $A$ and $B$ may be defined as the collection of all the elements which are present either in set $A$ or in set $B$.

In Set Builder Notation, the Union of two and Sets $A$ and $B$ may be written as,

## Symbol of Union of Two Sets

The Union of two sets $A$ and $B$ is denoted by $A ∪ B$.

## Explanation of Union of two sets

Let us consider two sets $A$ and $B$ such that,

$A=\{1, 2, 3, 4\}$

$B=\{4, 5, 6, 7\}$

The Union of the above two sets $A$ and $B$ will contain all the elements present in set $A$ and set $B$. The figure below shows the Venn Diagram for the Union of the two sets.

Therefore, the Union of $A$ and $B$ is given by,

$A ∪ B=\{1, 2, 3, 4, 5, 6, 7\}$

The element $4$ is common in both the sets, therefore, it is to be written only once after the union.

## Basic Properties of Union

### Cumulative Property of Union

The Cumulative Property of Union of two sets $A$ and $B$ is defined as:

$A ∪ B = B ∪ A$

#### Explanation of Cumulative Property of Union

To understand the Cummulative Property, let us consider two sets $A=\{1, 2, 3, 4\}$ and $B=\{4, 5, 6, 7\}$.

The Union of $A$ and $B$ is,

$A ∪ B=\{1, 2, 3, 4, 5, 6, 7\}$

and the Union of $B$ and $A$ is,

$B ∪ A=\{4, 5, 6, 7, 1, 2, 3\}$

From the above two results, it is observed that $A ∪ B$ and $B ∪ A$ has precisely the same elements, only the order of the elements is not the same. $B ∪ A$ can be rearranged as,

$B ∪ A =\{1, 2, 3, 4, 5, 6, 7\}$

Therefore, it can be concluded that,

$A ∪ B = B ∪ A$

This property of Union of two Sets is called the Cumulative Property of Union.

### Associative Property of Union

The Associative Property of Union is defined for three sets. The Associative Property of Union for three sets $A, B$ and $C$ may be defined as,

$A ∪ (B ∪ C) = (A ∪ B) ∪ C\,\,\,\,\,\,\,\,...........(i)$

#### Explanation of Associative Property of Union

To understand the concept of Associative Property of Union, let us consider three sets $A,\, B$ and $C$ such that,

$A = \{1, 2, 3\}$

$B = \{2, 3, 4\}$

$C = \{3, 4, 5\}$

First, find out the Left-Hand Side of the equation $(i)$,

$(B ∪ C) = \{2, 3, 4, 5\}$

$A ∪ (B ∪ C) = \{1, 2, 3\} ∪ \{2, 3, 4, 5\}$

$A ∪ (B ∪ C)= \{1, 2, 3, 4, 5\}$

Now find out the Right-Hand Side of equation $(i)$,

$(A ∪ B) = \{1, 2, 3, 4\}$

$(A ∪ B) ∪ C = \{1, 2, 3, 4\} ∪ \{3, 4, 5\}$

$(A ∪ B) ∪ C= \{1, 2, 3, 4, 5\}$

From the above two results it is observed that,

$LHS=RHS$

Therefore, it can be concluded that,

$A ∪ (B ∪ C) = (A ∪ B) ∪ C$

This Property of Union of two Sets is called the Associative Property of Union.

### Idempotent Property of Union

The Idempotent Property of Union states that the Union of a set $A$, with itself, is set $A$ only. Mathematically,

$A ∪ A = A$

#### Explanation of Idempotent Property of Union

To understand the Idempotent Property of Union, let us consider a set $A$ such that

$A = \{1, 2, 3, 4, 5\}$

Then, the Union of the set $A$ with itself will be,

$A ∪ A = \{1, 2, 3, 4, 5\} ∪ \{1, 2, 3, 4, 5\}$

$A ∪ A= \{1, 2, 3, 4, 5\}$

Therefore,

$A ∪ A= A$

This Property of Union is called the Idempotenet Property of Union.

### Identity Property of Union

The Identity Property of Union states that the Union of any set $A$ with an Empty Set is the set itself. Mathematically,

$A ∪ ∅ = A$

#### Explanation of Identity Property of Union

To understand the concept of Identity Property of Union, let us consider a set $A$ such that,

$A = \{1, 2, 3, 4, 5\}$

and an Empty set is given by,

$∅ = \{\}$

Therefore,

$A ∪ ∅ = \{1, 2, 3, 4, 5\} ∪ {}$

$A ∪ ∅= \{1, 2, 3, 4, 5\}$

$A ∪ ∅= A$

This Property of Union is called the Identity Property of Union.

## Solved Examples on Union of Sets

1. If $A=\{1, 2, 3, 4, 5\}$ and $B=\{a, c, f, g\}$, then find $A∪B$?

Solution: Given,
$A=\{1, 2, 3, 4, 5\}$
$B=\{a, c, f, g\}$
$A∪B$ will have all the elements present in the sets $A$ and $B$.
Therefore,
$A∪B=\{1, 2, 3, 4, 5, a, c, f, g\}$

2. If $A=\{a, b, c, d\}$, $B=\{1, 2, 3, 4\}$ and $C=\{x, y, z\}$, then prove that $A∪(B∪C)=(A∪B)∪C$?

Solution: Given,
$A=\{a, b, c, d\}$
$B=\{1, 2, 3, 4\}$
$C=\{x, y, z\}$
L.H.S:
$(B∪C)=\{1, 2, 3, 4, x, y, z\}$
Therefore, $A∪(B∪C)=\{a, b, c, d\}∪\{1, 2, 3, 4, x, y, z\}$
$A∪(B∪C)=\{a, b, c, d, 1, 2, 3, 4, x, y, z\}$
R.H.S:
$(A∪B)=\{a, b, c, d, 1, 2, 3, 4\}$
Therefore, $(A∪B)∪C=\{a, b, c, d, 1, 2, 3, 4\}∪\{x, y, z\}$
$(A∪B)∪C=\{a, b, c, d, 1, 2, 3, 4, x, y, z\}$
$L.H.S=R.H.S$
Therefore, $A∪(B∪C)=(A∪B)∪C$
Hence Proved..

3. If $A = \{ x|x \in N\}$ and $B = \{ x|x \in W\}$, then $A∪B=?$

Solution: Given,
$A = \{ x|x \in N\}$ which is the set of Natural Numbers.
$B = \{ x|x \in W\}$ is the set of Whole Numbers.
Therefore, the union of the two above sets $A$ and $B$ is the set of Whole Numbers.
$A∪B=\{ x|x \in W\}$

4. If $n(A)=10$ and $n(B)=5$ and there is no common elements in $A$ and $B$ then how many elements $A∪B$ will have?

Solution: Since there is no common elements between the sets $A$ and $B$, therefore,
$n(A∪B)=n(A)+n(B)$
$n(A∪B)=10+5$
$n(A∪B)=15$