#### 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.

- \( n(A)=p+s+v+u\)

This part is A. This parts represents the cardinality (or elements) which lies in A. - \( n(B)=q+s+v+t\)

This part is B. This parts represents the cardinality (or elements) which lies in B. - \( n(C)=r+u+v+t\)

This part is C. This parts represents the cardinality (or elements) which lies in C. - \( n(A \cap B)=s+v\)

This part is A and B. This parts represents the cardinality (or elements) which lies in \( A \cap B\). - \( n(A \cap C)=u+v\)

This part is A and C. This parts represents the cardinality (or elements) which lies in \( A \cap C\). - \( n(B \cap C)=t+v\)

This part is B and C. This parts represents the cardinality (or elements) which lies in \( B \cap C\). - \( n_o(A)=p\)

\(n(A-B-C)=p\)

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. - \( n_o(B)=q\)

\( n(B-C-A)=q\)

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. - \( n_o(C)=r\)

\(n(C-A-B)=r\)

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. - \( n_o(A \cap B)=s\)

\(n((A \cap B)-C)=s\)

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. - \( n_o(B \cap C)=t\)

\( n((B \cap C)-A)=t\)

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. - \( n_o(A \cap C)=u\)

\( n(A \cap C)-B)=u\)

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. - \(n(A \cap B \cap C)=v\)

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 - \(\overline{AUBUC}=w\)

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. - 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. - 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. - 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. - 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.

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