Parts of Three Set

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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.

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