Inscribed angle theorem

In geometry, a central angle is an angle whose vertex is at the center of a circle. A central angle is formed by two radii (plural of radius) of a circle. The central angle is equal to the measure of the intercepted arc. An intercepted arc is a portion of the circumference of a circle encased by two line segments meeting at the center of the circle

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Inscribed angle theorem

An inscribed angle in a circle is formed by two chords that have a common end point on the circle. This common end point is the vertex of the angle. In the figure below, circle with center O has the inscribed angle ∡ABC. The other end points than the vertex, A and C define the intercepted arc AC of the circle.

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Theorem

The measure of an inscribed angle is half the measure of the intercepted arc.
Proof
Given Consider a circle C with center O , we consider an inscribed angle at B by the arc AC
To Prove
∡B= \(\frac{1}{2} \measuredangle AOC\)
Construction
Join the vertices A and C with center O. Also draw a line through B and O .

SN	Statement	Reasons
1	∆BCO is Isosceles	CO=BO
2	y=2x	Triangle exteriar angle theorem
3	b=2a	Triangle exteriar angle theorem
4	y+b=2(a+x) a+x= \( \frac{1}{2}(y+b)\) ∡B= \(\frac{1}{2} \measuredangle AOC\)	Adding 2 and 3

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Symbolic Notation

Due to the theorem given above, it is seen that, the measure of arc AC has equal influence to the measure of its central angle ∡AOC. So it is also written as
\( \overset{⏜}{AC} \cong \measuredangle AOC \) or \( \overset{⏜}{AC} \equiv \measuredangle AOC \)
Similarly, the measure of chord AC has equal influence to the measure of its central angle ∡AOC. So it is also written as
\( \overline{AC} \cong \measuredangle AOC \) or \( \overline{AC} \equiv \measuredangle AOC \)
Similarly, the measure of chord AC has equal influence to the measure of its arc AC. So it is also written as
\( \overline{AC} \cong \overset{⏜}{AC} \) or \( \overline{AC} \equiv \overset{⏜}{AC} \)

Parts of Two Sets

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Parts of Two sets

Below is a Venn diagram involving two sets A and B

Here are four disjoint parts of the Venn-diagram. These four parts are

• \(A_o\) or \((A-B)\) [red color]
• \(B_o\) or \((B-A)\)[green Color]
• \(A \cap B\) [gray Color]
• \((AUB)'\) or \( (\overline{A \cup B})\) or \((AUB)^c\) [yellow color]

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Using these four disjoint parts, all together 16 different set notations can be formed. They are

1. 1 set notation can be formed taking 0 parts out of 4 disjoint parts
2. four different set notations can be formed taking 1 parts out of 4 disjoint parts
3. six different set notations can be formed taking 2 parts out of 4 disjoint parts
4. four different set notations can be formed taking 3 parts out of 4 disjoint parts
5. 1 set notation can be formed taking 4 parts out of 4 disjoint parts

Possible Combinations	Possible number of sets	Set Notations
Set with zero parts	1	\( \phi \)
Set with one parts	4	\(A_0,B_0,A \cap B, \overline{ (A \cup B)} \)
Set with two parts	6	\( A,B,\overline{A},\overline{B},A \triangle B, \overline{(A \triangle B)}\)
Set with three parts	4	\( \overline{(A-B)}, \overline{(B-A)},\overline{(A \cap B)}, A \cup B \)
Set with four parts	1	\( U \)

माथिको चारवटा अलगिएका समुहहरुलाई प्रयोग गरेर जम्मा 16 वटा फरक फरक समुहहरु बनाउन सकिन्छ । जसमा

1. 0 वटा भागलाई प्रयोग गरेर १ वटा समुह बनाउन सकिन्छ।
2. १ वटा भागलाई प्रयोग गरेर ४ वटा समुह बनाउन सकिन्छ।
3. २ वटा भागलाई प्रयोग गरेर ६ वटा समुह बनाउन सकिन्छ।
4. ३ वटा भागलाई प्रयोग गरेर ४ वटा समुह बनाउन सकिन्छ।
5. ४ वटा भागलाई प्रयोग गरेर १ वटा समुह बनाउन सकिन्छ।

These 16 different set notation are given below.

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1. Part 1: \(\phi\)

   Set Notation:\(\phi\)
   This part is formed taking 0 parts out of the four parts \(A_o,B_o,(A\cap B),(\overline{A \cup B})\)
   This part is also known as empty set.
   It contains no cardinality (or elements) of the sets A or B or U.
   समुह संकेत :\(\phi\)
   यो समुह बन्न को लागी दिएका चारवटा भागहरु \(A_o,B_o,(A\cap B),(\overline{A \cup B})\) मध्ये 0 वटा भाग को प्रयोग भएको छ।
   यसलाई खाली समुह पनि भनिन्छ ।
   यस समुहमा समुह A वा B वा U बाट कुनै पनि सदस्यहरु पर्दैनन् ।

2. Part 2: \(A_o\) or \(A-B \)

   Set Notation:\(A_o\) or \((A-B) \)
   This part is formed taking 1 part out of the four parts \(A_o,B_o,(A\cap B),(\overline{A \cup B})\)
   This part is also known as A difference B or only A
   It contains the cardinality (or elements) that belong to set A only, but not in B
   समुह संकेत :\(A_o\) or \(A-B \)
   यो समुह बन्न को लागी दिएका चारवटा भागहरु \(A_o,B_o,(A\cap B),(\overline{A \cup B})\) मध्ये १ वटा भाग को प्रयोग भएको छ।
   यसलाई A र B को फरक समुह भनिन्छ ।
   यसमा समुह A मा भएका तर B मा नभएका सदस्यहरु मात्र पर्दछन।

3. Part 3: B-A

   Set Notation:B-A (or B₀)
   This part is formed taking 1 parts out of the four parts A₀, B₀,A∩B and (AUB)'
   This part is also known as B difference A.
   It contains the cardinality (or elements) that belong to set B only, but not in A
   समुह संकेत :B-A (or B₀)
   यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये १ वटा भाग को प्रयोग भएको छ।
   यसलाई B र A को फरक समुह भनिन्छ ।
   यसमा समुह B मा भएका तर A मा नभएका सदस्यहरु पर्दछन।

4. Part 4: A∩B

   Set Notation:A∩B
   This part is formed taking 1 parts out of the four parts A₀, B₀,A∩B and (AUB)'
   This part is also known as A intersection B.
   It contains the cardinality (or elements) that belong to sets A and B both
   समुह संकेत :A∩B
   यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये १ वटा भाग को प्रयोग भएको छ।
   यसलाई A र B को प्रतिच्छेदन समुह भनिन्छ ।
   यसमा समुह A र B दुवैमा पर्ने साझा सदस्यहरु पर्दछन।

5. Part 5: (AUB)'

   Set Notation: (AUB)'
   This part is formed taking 1 parts out of the four parts A₀, B₀,A∩B and (AUB)'
   This part is also known as complement of A union B.
   It contains the cardinality (or elements) that belong to sets Neither A nor B nor both
   समुह संकेत : (AUB)'
   यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये १ वटा भाग को प्रयोग भएको छ।
   यसलाई A र B को संयोजन को पुरक समुह भनिन्छ ।
   यसमा समुह A वा B दुवैमा नपर्ने सदस्यहरु पर्दछन।

6. Part 6: A

   Set Notation: A
   This part is formed taking 2 parts out of the four parts A₀, B₀,A∩B and (AUB)'
   This part is also known as A .
   It contains the cardinality (or elements) that belong to sets A
   समुह संकेत :A
   यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये २ वटा भाग को प्रयोग भएको छ।
   यसलाई A समुह भनिन्छ ।
   यसमा समुह Aमा पर्ने सबै सदस्यहरु पर्दछन।

7. Part 7: B

   Set Notation:B
   This part is formed taking 2 parts out of the four parts A₀, B₀,A∩B and (AUB)'
   This part is also known as B .
   It contains the cardinality (or elements) that belong to sets B
   समुह संकेत :B
   यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये २ वटा भाग को प्रयोग भएको छ।
   यसलाई B समुह भनिन्छ ।
   यसमा समुह B मा पर्ने सबै सदस्यहरु पर्दछन।

8. Part 8: A'

   Set Notation:A'
   This part is formed taking 2 parts out of the four parts A₀, B₀,A∩B and (AUB)'
   This part is also known as complement of A .
   It contains the cardinality (or elements) that does NOT belong to set A
   समुह संकेत :A'
   यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये २ वटा भाग को प्रयोग भएको छ।
   यसलाई A को पुरक समुह भनिन्छ ।
   यसमा समुह A मा नपर्ने सबै सदस्यहरु पर्दछन।

9. Part 9: B'

   Set Notation:B'
   This part is formed taking 2 parts out of the four parts A₀, B₀,A∩B and (AUB)'
   This part is also known as complement of B .
   It contains the cardinality (or elements) that does NOT belong to set B
   समुह संकेत :B'
   यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये २ वटा भाग को प्रयोग भएको छ।
   यसलाई B को पुरक समुह भनिन्छ ।
   यसमा समुह B मा नपर्ने सबै सदस्यहरु पर्दछन।

10. Part 10: A∆B

    Set Notation:A∆B
    This part is formed taking 2 parts out of the four parts A₀, B₀,A∩B and (AUB)'
    This part is also known as union of (A-B) and (B-A) .
    It contains the cardinality (or elements) that belong to only one set
    समुह संकेत :A∆B
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये २ वटा भाग को प्रयोग भएको छ।
    यसलाई (A-B) र (B-A) को संयोजन समुह भनिन्छ ।
    यसमा समुह (A-B) वा (B-A) मा पर्ने सबै सदस्यहरु पर्दछन।

11. Part 11: (A∆B)'

    Set Notation: (A∆B)'
    This part is formed taking 2 parts out of the four parts A₀, B₀,A∩B and (AUB)'
    This part is also known as complement of the union of (A-B) and (B-A) .
    It contains the cardinality (or elements) that does NOT belong to only one set
    समुह संकेत : (A∆B)'
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये २ वटा भाग को प्रयोग भएको छ।
    यसलाई (A-B) र (B-A) को संयोजनको पुरक समुह भनिन्छ ।
    यसमा समुह (A-B) र (B-A) कुनैमा पनि नपर्ने सदस्यहरु पर्दछन।

12. Part 12: AUB

    Set Notation: AUB
    This part is formed taking 3 parts out of the four parts A₀, B₀,A∩B and (AUB)'
    This part is also known as union of A and B.
    It contains the cardinality (or elements) that belongs to either A or B or Both
    समुह संकेत : AUB
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये ३ वटा भाग को प्रयोग भएको छ।
    यसलाई A र B को संयोजन समुह भनिन्छ ।
    यसमा समुह A वा B मा पर्ने सदस्यहरु पर्दछन।

13. Part 13: \(A_0'\)

    Set Notation: (A₀)'
    This part is formed taking 3 parts out of the four parts A₀, B₀,A∩B and (AUB)'
    This part is also known as complement of A difference B.
    It contains the cardinality (or elements) that does Not belong to A only
    समुह संकेत : (A₀)'
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये ३ वटा भाग को प्रयोग भएको छ।
    यसलाई A र B को फरक को पुरक समुह भनिन्छ ।
    यसमा समुह A र B को फरकमा नपर्ने सबै सदस्यहरु पर्दछन।

14. Part 14: \(B_0'\)

    Set Notation: (B₀)'
    This part is formed taking 3 parts out of the four parts A₀, B₀,A∩B and (AUB)'
    This part is also known as complement of B difference A.
    It contains the cardinality (or elements) that does Not belong to B only
    समुह संकेत : (B₀)'
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये ३ वटा भाग को प्रयोग भएको छ।
    यसलाई B र A को फरक को पुरक समुह भनिन्छ ।
    यसमा समुह B र A को फरकमा नपर्ने सबै सदस्यहरु पर्दछन।

15. Part 15: (A∩B)'

    Set Notation: (A∩B)'
    This part is formed taking 3 parts out of the four parts A₀, B₀,A∩B and (AUB)'
    This part is also known as complement of A intersection B.
    It contains the cardinality (or elements) that does Not belong to both A and B
    समुह संकेत : (A∩B)'
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये ३ वटा भाग को प्रयोग भएको छ।
    यसलाई B र A को प्रतिच्छेदन को पुरक समुह भनिन्छ ।
    यसमा समुह A र B को प्रतिच्छेदनमा नपर्ने सबै सदस्यहरु पर्दछन।

16. Part 16: U

    Set Notation:U
    This part is formed taking 4 parts out of the four parts A₀, B₀,A∩B and (AUB)'
    This part is also known as full (Universal) set.
    It contains all cardinality (or elements) of the sets A or B or U.
    समुह संकेत :U
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये ४ वटा भाग को प्रयोग भएको छ।
    यसलाई सर्वव्यापक समुह भनिन्छ ।
    यसमा समुह A वा B वा U भएका सबै सदस्यहरु पर्दछन।

Operation on Sets

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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

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Union of Sets

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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}

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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 ।

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Intersection of Sets

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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}.

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}
   or A∩B ={4,5}
   or A∩B ={4,5}

   the shaded region is A∩B

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}
   or A∩B ={ }
   or A∩B ={ }

   the shaded region is A∩B , Empty Set

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}
   or A∩B ={4,5}
   or A∩B ={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}
   or A∩B ={4,5}
   or A∩B ={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}
   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 ।

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Difference of Sets

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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".

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

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

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}

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

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Complement of Set

Readings 👉 Click Here

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}.

1. 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}

2. 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}

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