1. if X is a set, find $\displaystyle X\cap P(X)$
I think it's $\displaystyle X$ but I don't know how to show it
2. if $\displaystyle P(X) = P(Y)$, is necessarily $\displaystyle X + Y$ justify your answer
$\displaystyle X = \left\{ {a,b} \right\}\,\& \,P(X) = \left\{ {\emptyset ,\{ a\} ,\{ b\} ,\{ a,b\} } \right\}$
Is there a single element in both of those sets?
As for #2. I have no idea what $\displaystyle X+Y$ could mean.
In the last century older mathematicians such as R L Moore used it to mean union. Is that the case here?