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Hello I am looking for help to understand how to go about the following.

Suppose we have some set B and are given that $$A \cup B=B \cap C$$

Then what would this tell us about B and C.

I tried to draw a venn diagram, but it didnt really make much sense.

I know that from definitions that the equality would mean that $$A \cup B \subset B \cap C$$ and that $$B \cap C \subset A \cup B$$

but im not to sure how to proceed? Could anyone help? Thanks

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It says first that $A\subset B$, since $A\subset A\cup B=B\cap C\subset B$. Hence $A\cup B=B=B\cap C$, which implies $B\subset C$. Summarising: $$A\subset B\subset C.$$ The converse is trivial.

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  • $\begingroup$ Thank you if possible could you just clarify exactly how we can conclude that B is subset of C from above? $\endgroup$ – PersonaA Sep 7 '15 at 20:56
  • $\begingroup$ Oh yes. It'd because the hypothesis says $A\cup B=B\cap C$, $B\cap C\subset C$, and we've just proved $A\cup B=B$. $\endgroup$ – Bernard Sep 7 '15 at 21:09
  • $\begingroup$ Thanks, and when it asks what I can say about C. Is there really anything that can be said about it other than what we have already? $\endgroup$ – PersonaA Sep 8 '15 at 0:38
  • $\begingroup$ No, nothing more really. Except, as the converse is trivially true, $A\cup B=A\cap C$ is equivalent to $A\subset B\subset C$. $\endgroup$ – Bernard Sep 8 '15 at 8:49
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$B\subset{A}\cup{B}={B}\cap{C}\subset{B}$

So $B=A\cup{B}=B\cap{C}$.

Thus, $A\subset{B}$ and $B\subset{C}$.

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  • $\begingroup$ Thank you. Can you possibly elaborate on how you can conclude the last line $\endgroup$ – PersonaA Sep 7 '15 at 18:44
  • $\begingroup$ Let $a\in{A}$. Then $a\in{A}\cup{B}=B$. Let $b\in{B}$. Then $b\in{B}\cap{C}\subset{C}$. $\endgroup$ – Hana Sep 8 '15 at 7:28

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