# Set Operation

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

1. $+$ to form a new number $5$ by $2+3$
2. $\times$ to form a new number $6$ by $2 \times 3$
3. we can do Set operation to form new Set by combining or manipulating one or more existing Sets.
4. Set operation helps to combine two or more sets together to form a new set.
5. 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}

#### 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}

#### 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}

#### 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}

#### 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}