## 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

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