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

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

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

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

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

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

Union of Sets (Definition, Examples, Symbol, Properties

Table of Contents

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,

A \cup B = \{ x: x \in A \text{ or } x \in B\}

Symbol of Union of Two Sets

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

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.

Union of Sets venn diagran — Union of 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:

$A \cup B = B \cup A$

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 $A \cup B$ and $B \cup A$ has precisely the same elements, only the order of the elements is not the same. $B \cup A$ can be rearranged as,

$B ∪ A =\{1, 2, 3, 4, 5, 6, 7\}$

Therefore, it can be concluded that,

$A \cup B = B \cup A$

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

Associative property of union venn diagram

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,

$A \cup (B \cup C) = (A \cup B) \cup C$

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,

$A \cup A = A$

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,

$A \cup A= A$

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,

$A \cup \emptyset = A$

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,

$A ∪ ∅ = \{1, 2, 3, 4, 5\} ∪ {}$

$A ∪ ∅= \{1, 2, 3, 4, 5\}$

$A \cup \emptyset= A$

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 $A\cupB$ ?

Solution: Given,
$A=\{1, 2, 3, 4, 5\}$
$B=\{a, c, f, g\}$
$A\cupB$ 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 $A\cup(B\cupC)=(A\cupB)\cupC$ ?

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, $A\cup(B\cupC)=(A\cupB)\cupC$
Hence Proved..

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

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 $A\cupB$ will have?

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

4.7/5 - (27 votes)

Abdur Rohman

Hi, my name is Abdur Rohman. By profession, I am an Electrical Engineer. I am also a part time Teacher, Blogger and Entrepreneur. The reason for starting this Website or Blog is mainly because I love teaching. Whenever I get time, I teach students/aspirants irrespective of their class or standards. More Info

Love this Post? Share with Friends