JEE Main & Advanced Mathematics Sets Some important results on number of elements in sets

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) \[n(A'\cap B')\text{ }=n(A\cap B)'=n(U)n(A\cup B)\]