(1) Prove or disprove: For any sets $X,Y,Z$ and any maps $f:X \to Y$ and $g:Y \to Z$, if $f$ is injective and $g$ is surjective, then $g \circ f$ is surjective.
So i proved previously that $f$ is injective if $g \circ f$ is injective, but then i also proved that if $g \circ f$ is surjective then $g$ is surjective. then this question appears, so how do i prove it that it is true or false? I'm saying it's true, because $g \circ f$ is bijective, so how do i go around proving it? Or my idea is already wrong? I need some help thanks.
(2) Show that (it is true in general that) for any sets $A,B$, one has $P(A) \cup P(B) \subseteq P(A \cup B)$.
I know that power set has exactly all the subsets of $A$ for $P(A)$. but I dont know how to start.