# Parts of Three Set

#### Parts of Three Sets

मानौ, सर्वव्यापक समुह U को उपसमुहहरु A,B र C छन भने तिन वटा समुहहरु समावेस भएका समस्याहरु समाधान गर्न तलको भेन चित्र प्रयोग गर्नुहोस। (Let A, B and C are the subsets of an universal set U, then use the following Venn-diagram to solve problems related to three sets.

1. $$n(A)=p+s+v+u$$
This part is A. This parts represents the cardinality (or elements) which lies in A.
2. $$n(B)=q+s+v+t$$
This part is B. This parts represents the cardinality (or elements) which lies in B.
3. $$n(C)=r+u+v+t$$
This part is C. This parts represents the cardinality (or elements) which lies in C.
4. $$n(A \cap B)=s+v$$
This part is A and B. This parts represents the cardinality (or elements) which lies in $$A \cap B$$.
5. $$n(A \cap C)=u+v$$
This part is A and C. This parts represents the cardinality (or elements) which lies in $$A \cap C$$.
6. $$n(B \cap C)=t+v$$
This part is B and C. This parts represents the cardinality (or elements) which lies in $$B \cap C$$.
7. $$n_o(A)=p$$
$$n(A-B-C)=p$$
This part is also known as A difference with B and C as denoted by A-B-C. This parts represents the cardinality (or elements) which lies in only in A but niether in B nor in C.
8. $$n_o(B)=q$$
$$n(B-C-A)=q$$
This part is also known as B difference with C and A as denoted by B-C-A. This parts represents the cardinality (or elements) which lies in only in B but niether in C nor in A.
9. $$n_o(C)=r$$
$$n(C-A-B)=r$$
This part is also known as C difference with A and B as denoted by C-A-B. This parts represents the cardinality (or elements) which lies in only in C but niether in B nor in A.
10. $$n_o(A \cap B)=s$$
$$n((A \cap B)-C)=s$$
This part is also known as intersection of A and B, only. This parts represents the cardinality (or elements) which lies in only intersection of A and B but NOt in C.
11. $$n_o(B \cap C)=t$$
$$n((B \cap C)-A)=t$$
This part is also known as intersection of B and C, only. This parts represents the cardinality (or elements) which lies in only intersection of B and C but NOT in A.
12. $$n_o(A \cap C)=u$$
$$n(A \cap C)-B)=u$$
This part is also known as intersection of A and C, only. This parts represents the cardinality (or elements) which lies in only intersection of A and C but NOT in B.
13. $$n(A \cap B \cap C)=v$$
This part is also known as intersection of A , B and C. This parts represents the cardinality (or elements) which lies in A, B and C, in all three sets.
At least three sets
Exactly three sets
14. $$\overline{AUBUC}=w$$
This part is also known as complement of union of A , B and C. It is also denoted by $$(A \cup B \cup C)'$$ or $$(A \cup B \cup C)^c$$. This parts represents the cardinality (or elements) which does NOT lier on either A or B or C.
15. Only one: $$n(A_0)+n(B_0)+n(C_0)=p+q+r$$
This part is known as "Like only one" . This parts represents the cardinality (or elements) which lies on Exactly one.
16. Only two: $$n_0(A \cap B)+n_0(B \cap C)+n_0(A \cap C)=s+t+u$$
This part is known as "Like only two" . This parts represents the cardinality (or elements) which lies on Exactly two.
17. At least one: $$n(A \cup B \cup B)=p+q+r+s+t+u+v$$
This part is known as "at least one" . This parts represents the cardinality (or elements) which lies on either A or B or C.
18. At least two: $$n_0(A \cap B)+n_0(B \cap C)+n_0(A \cap C)+n( A\cap B \cap C) =s+t+u+v$$
This part is known as "at least two" . This parts represents the cardinality (or elements) which lies at least two sets.