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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.

Andrés E. Caicedo
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user137243
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  • Yes. It is. Adding more characters so I can post comment. –  Apr 17 '14 at 01:55
  • Do you mean I should add more characters on my post so you can post comments? – user137243 Apr 17 '14 at 01:56
  • No I mean that I had to add characters to post my comment. :) Try posting a comment with one word it won't work. –  Apr 17 '14 at 01:57
  • Probably you need to prove the transitive property. Or have you already seen a proof of it? – Andrés E. Caicedo Apr 17 '14 at 01:57
  • No I have not. That's why i asked because I feel that simply stating its transitive property is not really a proof – user137243 Apr 17 '14 at 01:59
  • @user137243: Can you tell me where this problem is taken from? (I don't care if it's hw.) – freishahiri Jul 20 '15 at 11:04
  • @NotNotLogical: You can achieve the character minimum with invisible characters by adding something like this to a comment: ${}{}{}{}{}$ For example, see this comment. Also, I'm not sure who downvoted this question. It seems perfectly clear and reasonable to me. I have upvoted. – Cameron Buie Aug 26 '15 at 11:55
  • Duplicate of [this](http://math.stackexchange.com/questions/15188/how-to-prove-a-subset-b-b-subset-c-c-subset-a-rightarrow-b-c? That question was shown in the sidebar among the related questions. – Martin Sleziak Sep 30 '15 at 14:09

2 Answers2

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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!)

freishahiri
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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 $B=C$.

mjh
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