[Math] Let $A,B$, and $C$ be sets. If $A\subseteq B$, $B\subseteq C$, and $C\subseteq A$, then $A=C$.

elementary-set-theoryproof-writing

Here is my abstract maths problem.

Let $A,B$, and $C$ be sets. If $A\subseteq B$, $B\subseteq C$, and $C\subseteq A$, then $A=C$.

I am asked to either prove or disprove this statement. I am a little confused about the best way to form this proof. Here is my thought process thus far:

We know that:

$A \subseteq B \Rightarrow \forall \ x \in A, \ x \in B $

$B \subseteq C \Rightarrow \forall \ x \in B, \ x \in C $

$C \subseteq A \Rightarrow \forall \ x \in C, \ x \in A $

Provided this this information, I can show that $A=C$ by showing that $A \subseteq C$ and $C \subseteq A$.

From this point, I guess I'm not sure which directionality to show. Can I use a direct proof in the form of "If $A=C$, then $A\subseteq B$, $B\subseteq C$, and $C\subseteq A$." I don't think that's allowed (right?), so would it then be best to do this proof in the form of a contradiction using the converse direction?

Thanks for looking. I think I know how to prove this, but I just don't know to what method I should format my proof.

Mia

Best Answer

HINT: No, you can’t assume that $A=C$ and show that the inclusions hold: that’s the converse of what you’re supposed to prove, and an implication and its converse are not logically equivalent.

  • Use the fact that $A\subseteq B$ and $B\subseteq C$ to show that $A\subseteq C$.

You’re given that $C\subseteq A$, so the rest is straightforward.

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