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To prove A=B, I must prove that A is a subset of B and B is a subset of A. A is a subset of B is already given. So all that is left is to prove B is a subset of A.

Is it suffice to say that since A is a subset of B, B is a subset of C, and C is a subset of A, by transitive property B is a subset of A.

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  • $\begingroup$ Yes. It is. Adding more characters so I can post comment. $\endgroup$
    – user142299
    Apr 17, 2014 at 1:55
  • $\begingroup$ Do you mean I should add more characters on my post so you can post comments? $\endgroup$
    – user137243
    Apr 17, 2014 at 1:56
  • $\begingroup$ No I mean that I had to add characters to post my comment. :) Try posting a comment with one word it won't work. $\endgroup$
    – user142299
    Apr 17, 2014 at 1:57
  • $\begingroup$ Probably you need to prove the transitive property. Or have you already seen a proof of it? $\endgroup$ Apr 17, 2014 at 1:57
  • $\begingroup$ No I have not. That's why i asked because I feel that simply stating its transitive property is not really a proof $\endgroup$
    – user137243
    Apr 17, 2014 at 1:59

2 Answers 2

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In fact, the problem deals wit two separate issues!

Check transitivity: $$A\subseteq B\subseteq C\implies A\subseteq C$$

Check antisymmetry: $$A\subseteq C\subseteq A\implies A=C$$

(These are rather different!)

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If you are given that $A\subseteq B\subseteq C\subseteq A$, then yes you can. To be more specific, given any $x\in B$, $x$ is also in $C$, and also in $A$, so $B\subseteq A$. Use a simlar argument to show that $B=C$.

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