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Let $M,C,P,F,$ be nonempty sets satisfying the following conditions:

  1. $M\subset C$;
  2. $M\cap P\neq \emptyset$;
  3. $C\cap F\neq \emptyset$;
  4. $F\subset C\cup P$;
  5. $P\cap C^{c}\neq \emptyset$.

Is it true that $F\subset M\cup P?$ I was told by a friend of mine that it is true.

I wasn't able to solve that. If I start by saying that if $ x\in F $, then by (4) I get $x\in C$ or $x\in P$ and I got stuck. Then I'tried another way. If $x\notin M\cup P$ then I get $x\notin M$ or $x\notin P$, but again, I don't know how use all the hypothesis.

I would appreciate your help.

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up vote 6 down vote accepted

It’s not necessarily true. Let $M=\{1\}$, $C=\{1,2\}$, $P=\{1,3\}$, $F=\{2\}$; all five conditions are satisfied, but $F\cap(M\cup P)=\varnothing$.

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