Can you help me with the following exercise? The main reason I can't do it is because I think it's impossible.
Given A and B sets, let X be a set with the following properties:
P1) $X\supset A$ and $X\supset B$
P2) If $Y\supset A$ and $Y\supset B$ then $Y\supset X$
Prove that $X=A\cup B$
From the way I see it, if X has the properties P1 and P2, $A\cup B \subset X$, but not necessarily $X \subset A \cup B$. That is, I think the properties mean X will contain $A\cup B$ but X can be much bigger than that. I don't see how $X\setminus A\cup B$ is necessarily empty. I don't understand the use of P2, either. How does P2 constrain X to exactly $A\cup B$?
Thus, I don't think I can prove what it asks because it's wrong. But I feel I'm missing something. Any help is appreciated.