Set Operation

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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}

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?

Similary, try yourself to understand the conceept of intersection, difference and complement

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