Are projections surjective? Suppose $A,B$ are non empty and consider projection $P: A \times B \to A $ given by $P( (a,b) ) = a $. Show $P$ is surjective.
Attempt:
Let $y \in A $ be arbitrary. We know that for every $x \in B $, it follows that $P ( ( y,x) ) = y $. So, for every $y \in A $, we can always find an element $(a,b) \in A \times B $, namely $(a,b) = (y,x) $ so that $P(y,x) = y $. In fact, $P$ is surjective.
Is this correct?

Pretty much yes. You need to cite non-emptiness of $B$ explicitly... but I would still give full credit.
