In real number system, we can do four fundamental operation to form new number by combining or manipulating one or more existing numbers. For example, given two numbers \(2\) and \(3\) , we can use

- \(+\) to form a new number \(5\) by \(2+3\)
- \(\times\) to form a new number \(6\) by \(2 \times 3\)
- we can do Set operation to form new Set by combining or manipulating one or more existing Sets.
- Set operation helps to combine two or more sets together to form a new set.
- The common example of set operations are: Union, Intersection, Difference, and Complement

#### Union of Sets

Let A and B be any two sets. Then union of sets A and B is a new set consisting all the elements of A and B without repetition. The union is the smallest set containing elements of A and B.

It is denoted by AUB and read as “A union B” or “A cup
B”.

Mathematically,

AUB = {x: x ∈ A or x ∈ B}.

मानौ A र B कुनै दुई समुहहरू छन । अब समुह A र B को संयोजन (union) भनेको एउटा नयाँ समुह हो जुन A र B का सबै सदस्यहरु समावेश भई बनेको हुन्छ। संयोजन समुह A र B बाट बन्ने सबैभन्दा सानो समुह हो । यसलाई AUB ले जनाईन्छ र "A संयोजन B" भनेर पढिन्छ।

गणितिय भाषामा,

AUB = {x: x ∈ A or x ∈ B}.

#### Example 1

If A={ 1,2,3,4,5} and B={4,5,6,7,8}, then find A∪B

Solution

In this example, A={ 1,2,3,4,5} and B={4,5,6,7,8}

Thus,

A∪B={Common Elements of A and B} ∪ {Remaining element of A} ∪ {Remaining element of B}

or A∪B={4,5} ∪{1,2,3}∪{6,7,8}

or A∪B={1,2,3,4,5,6,7,8}

the shaded region is A∪B

#### Example 2

If A={ 1,2,3} and B={6,7,8}, then find A∪B

Solution

In this example, A={ 1,2,3} and B={6,7,8}

Thus,

A∪B={Common Elements of A and B} ∪ {Remaining element of A} ∪ {Remaining element of B}

or A∪B={ }∪{1,2,3}∪{6,7,8}

or A∪B={1,2,3,6,7,8}

the shaded region is A∪B

#### Example 3

If A={ 1,2,3,4,5} and B={4,5}, then find A∪B

Solution

In this example, A={1,2,3,4,5} and B={4,5}

Thus,

A∪B={Common Elements of A and B} ∪ {Remaining element of A} ∪ {Remaining element of B}

or A∪B={4,5} ∪{1,2,3}∪{}

or A∪B={1,2,3,4,5}

the shaded region is A∪B

#### Example 4

If B={ 1,2,3,4,5} and A={4,5}, then find A∪B

Solution

In this example, B={1,2,3,4,5} and A={4,5}

Thus,

A∪B={Common Elements of A and B} ∪ {Remaining element of A} ∪ {Remaining element of B}

or A∪B={4,5} ∪{1,2,3}∪{}

or A∪B={1,2,3,4,5}

the shaded region is A∪B

#### Example 5

If A={1,2,3,4,5} and B={1,2,3,4,5}, then find A∪B

Solution

In this example, A={1,2,3,4,5} and B={1,2,3,4,5}

Thus,

A∪B={Common Elements of A and B} ∪ {Remaining element of A} ∪ {Remaining element of B}

or A∪B={1,2,3,4,5} ∪{}∪{}

or A∪B={1,2,3,4,5}

the shaded region is A∪B

#### Question

How many different Venn-Diagram can be formed for union of two sets?

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