# An exercise on set theory

The type of exercise I'm about to speak of seems quite basic however I didn't have any exposure to it until recently so please provide me with some pointers on how to work it out.

Show that if the sets A, B and C satisfy the following relations simultaneously:

$A\cup B = C\\ (A\cup C) \cap B = C\\ (A \cap C) \cup B = A$

then they are the same.

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1
For example, it is possible to read $B\subseteq A\subseteq C\subseteq B$ from the given statements (how?), hence equality must hold. – Hagen von Eitzen Oct 7 '12 at 14:57

## 2 Answers

Recall that $X=Y$ if and only if $X\subseteq Y$ and $Y\subseteq X$.

Now using this we have that: $B\subseteq A\cup B\subseteq C$ from the first line; and since $C=B\cap (\text{something})$ we have that $C\subseteq B$. From those we have that $B=C$.

I am leaving it to you to show how $A=B$ as well, and to write a rigorous proof.

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Thanks to Asaf Karagila's answer and my last Logics course I understood how to solve this type of exercise. I have attached my solution down below in a picture however I intend to rewrite this answer using latex equations when I'll get some free time.

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