# Parts of Sets

अंकगणितमा +,-,x, ÷ लाई चारवटा आधारभतु क्रियाहरु मानिन्छ । In school mathematics, यि चारवटा आधारभतु क्रियाहरु बाहेक "Set" लाई पनि counting principle को रुपमा आधारभुत क्रिया भनिन्छ । भेनडायग्राम को प्रयोग गरेर समुहबाट counting system सम्बन्धी समस्याहरुको समाधान गर्न सकिन्छ । जसलाई big ideas पनि भनिन्छ।

दुईवटा समुहहरु सबममबलत भएको िाबददक समस्याहरु बिक्षि को योजना
1. Testing the set operation
2. Testing the Cardinality
3. Parts of the sets (4 disjoint pieces and 16 possibilities)
4. Arithmetic’s on Cardinality
5. Identification of set notation
6. Problem solving (Context specific)

## Parts of the sets

Below is a Venn diagram involving two sets A and B

We can make four disjoint parts of the above Venn-diagram, which are as below.

1. ##### Part 1: A-B

This part is known as A difference B
It is denoted by A−B
It is also denoted by Ao
It represents the cardinality (or elements) which lies in only A but not in B

2. ##### Part 2: A∩B

This part is also known as A intersection B
This part is denoted by A∩B
This parts represents the cardinality (or elements) which lies both in A and B.

3. ##### Part 3: B-A

This part is also known as B difference A
This part is denoted by B-A
This part is also denoted by Bo
This parts represents the cardinality (or elements) which lies in only B but not in A

4. ##### Part 4: (AUB)'

This part is also known complement of A union B
This parts represents the cardinality (or elements) which lies neither in A nor in B

#### Different notion of sets using four disjoint parts

Below is a Venn diagram involving two sets A and B

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

• A0 (or A-B)[red color]
• B0 (or B-A)[green Color]
• A∩B [gray Color]
• (AUB)' [pink color]

Now,how many different sets can be formed using these four disjoint parts.
Using these four disjoint parts, all together 16 different set notation can be formed. More explicitely

1. 1 set notation can be formed taking 0 parts out of 4 disjoint parts
2. four different set notation can be formed taking 1 parts out of 4 disjoint parts
3. six different set notation can be formed taking 2 parts out of 4 disjoint parts
4. four different set notation 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 $A_0,B_0,A \cap B, (A \cup B)'$ Set with two parts 6 $A,B,A',B,A \triangle B, (A \triangle B)'$ Set with three parts 4 $(A-B)', (B-A)',(A \cap B)', A \cup B$ Set with four parts 1 $U$

अब, माथिको भेन चित्रको आधारमा चारवटा अलगिएका समुहहरुलाई प्रयोग गरेर कतिवटा फरक फरक समुहहरु बनाउन सकिन्छ?
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।

Therefore, all together 16 different set notation can be formed.
These 16 different set notation are given below.

1. ##### Part 1: 𝜙

Set Notation:𝜙
This part is formed taking 0 parts out of the four parts A0, B0,A∩B and (AUB)'
This part is also known as empty set.
It contains no cardinality (or elements) of the sets A or B or U.
समुह संकेत :𝜙
यो समुह बन्न को लागी दिएका चारवटा भागहरु A0, B0,A∩B र (AUB)'मध्ये 0 वटा भाग को प्रयोग भएको छ।
यसलाई खाली समुह पनि भनिन्छ ।
यस समुहमा समुह A वा B वा U कुनैमा पनि नभएका सदस्यहरु पर्दछन।

2. ##### Part 2: A-B

Set Notation:A-B (or A0)
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 difference B.
It contains the cardinality (or elements) that belong to set A only, but not in B
समुह संकेत :A-B (or A0)
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 फरक समुह भनिन्छ ।
यसमा समुह A मा भएका तर B मा नभएका सदस्यहरु 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।

##### Arithemetic of Cardinality

Arithmetic of cardinality in sets refers to the mathematical operations that involve counting the number of elements (cardinality) within sets. When solving verbal problems involving sets, you might encounter situations where you need to perform arithmetic operations such as addition, subtraction, multiplication, and division on the cardinalities of sets to find the desired information.
Here are some common scenarios where arithmetic of cardinality comes into play when solving verbal problems related to sets:

1. Union of Sets: When we need to find the total number of elements in the union of two or more sets, we use the concept of cardinality. For example, if we have sets A and B, the cardinality of their union (A ∪ B) can be calculated by adding the cardinalities of A and B and then subtracting the cardinality of their intersection (A ∩ B) to avoid double counting any shared elements.
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
2. Complements: The complement of a set A with respect to a larger set U (universal set) contains all elements in U that are not in A. You can calculate the cardinality of the complement by subtracting the cardinality of A from the cardinality of U.
n(A') = n(U) - n(A)
3. Subtraction of Sets: When you want to find the number of elements in one set that are not in another set, you can use subtraction of cardinalities. For instance, if you have sets A and B, the cardinality of the difference A - B is found by subtracting the cardinality of B from the cardinality of A.
n(A - B) = n(A) - n(B)
1. #### Test your Understandings: n(AUB)

From the diagram below, find the cardinal number of AUB.

1. $n(A \triangle B)$

2. $n(A-B)$

3. $n(B-A)$

4. $n(A \cap B)$

5. $n(A \cup B)'$

6. $n(A \cup B)$

7. $n(A)$

8. $n(B)$

9. $n(A)'$

10. $n(B)'$