Operation on Sets

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1. Introduction 2. Union of Sets 3. Intersection of Sets 4. Difference of Sets 5. Complement of Sets 6. Test

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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

Union of two 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.
In other words
The union of two sets A and B is the set of elements which are in A, in B, or in both 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}.

1. 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}

   A∪B by shaded region

2. 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

3. 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

4. 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

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}

Union of Three Sets

Let A, B and C be any three sets. Then union of sets A, B and C is a new set consisting all the elements of A, B and C without repetition. The union is the smallest set containing elements of A, B and C.
In other words
The union of three sets A, B and C is the set of elements which are in A, in B, in C or in both A, B and C
It is denoted by AUBUC and read as “A union B union C” or “A cup B cup C”.
Mathematically,
AUBUC = {x: x ∈ A or x ∈ B or x ∈ C}.

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

गणितिय भाषामा,
AUBUC = {x: x ∈ A or x ∈ B or x ∈ C}.
समूहको संयोजन गर्दा दिइएका समूहका साझा सदस्यहरूलाई नदोहो-याइकन बाँकी सबै सदस्यहरूलाई लिएर समूहको रूपमा लेख्नुपर्छ ।

Example 1

If U={a, b, c, d, e,f,g,h,i,o,u}, A = {a, b, c, d, e}, B = {a, e, i, o, u}, C = {d, e, f, g} are given then find \(A \cup B \cup C\) and present it in Venn-Diagram.
Given that
U={a, b, c, d, e,f,g,h,i,o,u}
A = {a, b, c, d, e}
B = {a, e, i, o, u}
C = {d, e, f, g}
The union of A,B and C is given by
AUBUC = {x: x ∈ A or x ∈ B or x ∈ C}.
or AUBUC = {a, b, c, d, e,f,g,i,o,u}
सँगैको भेनचित्रमा छाया पारेको भागले AUBUC लाई जनाउँछ ।

Intersection of Sets

Let A and B be any two sets. Then intersection of sets A and B is a new set consisting common elements of A and B. The intersection is the largest set containing common elements of A and B.
It is denoted by A∩B and read as “A intersection B” or “A cap B”.
Mathematically, A∩B = {x: x ∈ A and x ∈ B}.

मानौ A र B कुनै दुई समुहहरू छन । अब समुह A र B को प्रतिच्छेदन (intersection) भनेको एउटा नयाँ समुह हो जुन A र B का सबै साझा सदस्यहरु समावेश भई बनेको हुन्छ। प्रतिच्छेदन समुह A र B को साझा सदस्यबाट बन्ने सबैभन्दा ठुलो समुह हो । यसलाई A∩B ले जनाईन्छ र "A प्रतिच्छेदन B" भनेर पढिन्छ।

गणितिय भाषामा, A∩B = {x: x ∈A and x ∈ B}.

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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}
or A∩B ={4,5}

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}
or A∩B ={ }
or A∩B ={ }

the shaded region is A∩B , Empty Set

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}
or A∩B ={4,5}
or A∩B ={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}
or A∩B ={4,5}
or A∩B ={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}
or A∩B ={1,2,3,4,5}
or A∩B ={1,2,3,4,5}

the shaded region is A∩B

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Intersection of Three Sets

Let A, B and C be any three sets. Then intersection of sets A, B and C is a new set consisting all the COMMON elements of A, B and C without repetition. The union is the largest set containing the COMMON elements of A, B and C.
In other words
The intersection of three sets A, B and C is the set of elements which are in A, and in B, and in C
It is denoted by A∩B∩C and read as “A intersection B intersection C” or “A cap B cap C”.
Mathematically,
A∩B∩C = {x: x ∈ A and x ∈ B and x ∈ C}.

मानौ A, B र C कुनै तिन समुहहरू छन । अब समुह A, B र C को प्रतिच्छेदन (intersection) भनेको एउटा नयाँ समुह हो जुन A, B र C का सबै साझा सदस्यहरु समावेश भई बनेको हुन्छ। प्रतिच्छेदन समुह A, B र C को साझा सदस्यबाट बन्ने सबैभन्दा ठुलो समुह हो । यसलाई A∩B∩C ले जनाईन्छ र "A प्रतिच्छेदन B प्रतिच्छेदन C " भनेर पढिन्छ।

गणितिय भाषामा,
A∩B∩C = {x: x ∈ A and x ∈ B and x ∈ C}.
समूहको प्रतिच्छेदन गर्दा दिइएका सबै समूहका साझा सदस्यहरूलाई मात्र नदोहो-याइकन समूहको रूपमा लेख्नुपर्छ ।

Example 1

If U={a, b, c, d, e,f,g,h,i,o,u}, A = {a, b, c, d, e}, B = {a, e, i, o, u}, C = {d, e, f, g} are given then find \(A \cap B \cap C\) and present it in Venn-Diagram.
Given that
U={a, b, c, d, e,f,g,h,i,o,u}
A = {a, b, c, d, e}
B = {a, e, i, o, u}
C = {d, e, f, g}
The union of A,B and C is given by
A∩B∩C = {x: x ∈ A and x ∈ B and x ∈ C}.
or A∩B∩C = {a, b, c, d, e}∩{a, e, i, o, u}∩{d, e, f, g}
or A∩B∩C = {e}
सँगैको भेनचित्रमा घेरा पारेको भागले A∩B∩C लाई जनाउँछ ।

Difference of Sets

Let A and B be any two sets. Then difference of sets A and B is a new set consisting elements of only A which are NOT in B.
It is denoted by A-B and read as “A difference B” or “A - B”.
Mathematically, A-B = {x: x ∈ A and x ∉ B}.

मानौ A र B कुनै दुई समुहहरू छन । अब समुह A र B को फरक (difference) भनेको एउटा नयाँ समुह हो जुन A मा मात्र भएको तर B मा नभएको सबै सदस्यहरु समावेश भई बनेको हुन्छ। यसलाई A-B ले जनाईन्छ र "A फरक B" भनेर पढिन्छ।

गणितिय भाषामा, A-B = {x: x ∈A and x ∉ B}.
The union of A-B and B-A is called symmetric difference of A and B, and it is denoted by \(A \triangle B\) or \(A \ominus B\), and read as " A symmetric difference 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 =Elements in Abut NOT in B
or A-B ={1,2,3}
or A-B ={1,2,3}

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 =Elements in Abut NOT in B
or A-B ={1,2,3}
or A-B ={1,2,3}

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 =Elements in Abut NOT in B
or A-B ={1,2,3}
or A-B ={1,2,3}

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 =Elements in Abut NOT in B
or A-B ={}
or 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 =Elements in Abut NOT in B
or A-B ={}
or A-B ={}

Complement of Set

Let A and B be any two sets. Then Complement of sets A is a new set consisting elements which are NOT in A.
It is denoted by A' or \(\overline{A}\) and read as “A Complement” or “U - A”.
Mathematically, A' = {x: x ∈ U and x ∉ A}.

मानौ A कुनै एउटा समुह हो । अब समुह A को पुरक (Complement) भनेको एउटा नयाँ समुह हो जुन A मा नभएको सबै सदस्यहरु समावेश भई बनेको हुन्छ। यसलाई A' or \(\overline{A}\) ले जनाईन्छ र "U-A" भनेर पढिन्छ।

गणितिय भाषामा, A' = {x: x ∈ U and x ∉ A}.

Example 1

If U={ 1,2,3,4,5} and A={4,5} , then find A'
Solution
In this example,
U={ 1,2,3,4,5} and A={4,5}
Therefore, A'= U-A={1,2,3}

Example 1

If U={ 1,2,3,4,5,6,7,8,9,10} with A={1,2,3,4,5}, B={4,5,6,7,8} , then find A'
Solution
In this example,
A'= U-A={6,7,8,9,10}