Some important results on number of elements in sets
Category : JEE Main & Advanced
If A, B and C are finite sets and U be the finite universal set, then
(1) \[n(A\cup B)=n(A)+n(B)-n(A\cap B)\]
(2) \[n(A\cup B)=n(A)+n(B)\Leftrightarrow A,\,\,B\] are disjoint non-void sets.
(3) \[n(A-B)=n(A)-n(A\cap B)\] i.e., \[n(A-B)+n(A\cap B)=n(A)\]
(4) \[n(A\Delta B)=\] Number of elements which belong to exactly one of A or B \[=n((A-B)\cup (B-A))=n(A-B)+n(B-A)\]
\[[\because \,\,(A-B)\] and \[(B-A)\] are disjoint]
\[=n(A)n(A\cap B)+n(B)n(A\cap B)=n(A)+n(B)2n(A\cap B)\]
(5) \[n(A\cup B\cup C)=n(A)+n(B)+n(C)n(A\cap B)n(B\cap C)n(A\cap C)+n(A\cap B\cap C)\]
(6) n (Number of elements in exactly two of the sets A, B, C) \[=n(A\cap B)+n(B\cap C)+n(C\cap A)3n(A\cap B\cap C)\]
(7) n(Number of elements in exactly one of the sets A, B, C) \[=n(A)+n(B)+n(C)\]
\[2n(A\cap B)2n(B\cap C)2n(A\cap C)+3n(A\cap B\cap C)\]
(8) \[n(A'\cap B')\text{ }=n(A\cap B)'=n(U)n(A\cap B)\]
(9) \[n(A'\cap B')\text{ }=n(A\cap B)'=n(U)n(A\cup B)\]
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