I am trying to get an intuitive understanding of how reflexivity proofs like these work.
Suppose R is a relation on A, and define a relation S on P(A) as follows:
$S=\{ (X,Y)\in P(A)\times P(A) |\forall x\in X \exists y\in Y (xRy)\} $
Show that S is reflexive.
(It's not reflexive; but it doesn't matter, it only serves as an example)
In proofs like these, one often begins by letting an element of the domain as arbitrary, in this case - letting $X$ as an arbitrary element of $P(A)$. Then one would proceed to utilise other assumptions in combination of this arbitrary element.
It is the legitimacy of this arbitrary element assumption that I am not sure of. Why are we justified to assume an arbitrary element like this? This step is identical to what one would do in a universal proof, but this looks nothing like a universal proof.
Is it because, since we have defined $S$ like this, the definition of $S$ itself has already implicitly assumed that $P(A)$ is not empty (otherwise it would be vacuous)?
I am a self-teaching student so I am really grateful for any help on this!
What you are trying to do is to show that $zSz$ holds for any $s \in P(A)$. To do that, you must show either that (1) $P(A) = \emptyset$ (which is not true here since $P(A)$ is a power set) or that (2) any element $z$ in $P(A)$ satisfies $zSz$. So selecting an arbitrary element and showing $zSz$ holds establishes this is true for all elements of $P(A)$...