Properties of Cardinality and intersection I have three sets A, B and C satisfying the following conditions: * $ \\#(A\cap B) = 11$ * $ \\#(A\cap C) = 12$ * $ \\#(A\cap B\cap C) = 5$ What is the minimun cardinality of A? What I did was this: (11 - 5) + (12 - 5) = 13. But I'm not sure if I have to substract the 5 twice or only once. And the fact that I don't know if B and C are disjoint gets me confused =/

\begin{align} \\#(A\cap (B\cup C)) &= \\#((A\cap B)\cup (A\cap C))\\\ &=\\#(A\cap B)+\\#(A\cap C)-\\#(A\cap B\cap C)\\\ &=11+12-5=18. \end{align} Note that $A$ cannot be fewer than $18$ since $A\cap(B\cup C)\subseteq A$, thus $\\#(A)\geq18$.

Now, if we can find an example, where $\\#(A)=18$, we'll have minimized the cardinality of $A$. For such an example, let $A=\\{1,\ldots,18\\}$, $B=\\{1,\ldots,11\\}$, and $C=\\{7,\ldots,18\\}$.