If $A$ is a subset of $B$, $B$ is a subset of $C$ and $C$ is a subset of $A$, prove that $A=C$

Question

Please help I didn't get any idea to solve.

Well, how do you define "subset"? How do you prove that two sets are equal? — martin.koeberl, Mar 28 '17 at 14:03
You have to "cook together" two definitions: that of (set) inclusion : $\subseteq$ and that of (set) equality : $=$. Both are expressed in terms of the "basic" set relation : $\in$. — Mauro ALLEGRANZA, Mar 28 '17 at 14:19
Possible duplicate of prove that if A is a subset of B, B is a subset of C, and C is a subset of A, then A=B and B=C — Mithlesh Upadhyay, Mar 28 '17 at 15:13

score 2 · Accepted Answer · answered Mar 28 '17 at 14:03

2

Hints:

answered Mar 28 '17 at 14:03

drhab

what is the hint here? – Aryabhatta Mar 28 '17 at 14:06
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Apply both hints. E.g. what can you conclude from $A\subseteq B\wedge B\subseteq C$ if $\subseteq$ is transitive? And what can you conclude from $A\subseteq C\wedge C\subseteq A$ if $\subseteq$ is antisymmetric? – drhab Mar 28 '17 at 14:08
I did not understand. Could you please elaborate? – Aryabhatta Mar 28 '17 at 14:09
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What is your answer to both questions in my former comment? – drhab Mar 28 '17 at 14:10

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