Ans (C).
$\left(P \rightarrow Q\right) \leftrightarrow \left(\neg P \vee Q\right)$
(D) is wrong as shown below.
Let $S = \left\{2, 3, 4, 5\right\}$ and $P(x,y)$ be $x < y$.
Now, $P(2,3)$ is true but $P(3,2), P(4,2)$ etc are false and hence the implication also.
This is because the given formula is evaluated as:
$\forall x \, \forall y \, \left(P(x,y) \, \rightarrow \, \forall x \, \forall y \, P(y, x) \right)$
For every (x,y) if P(x,y) is true then for every (x,y) P(y,x) is true.
On the RHS, $P(y,x)$ can be replaced with $P(x,y)$ and then also the formula means the same. So, here precedence rule used is $\implies$ having more precedence than quantification which is against the convention used in Wikipedia. I guess all books only talk about conventions and there is no standard here. (C) option being so straight forward I guess, GATE didnot even consider this as an ambiguity. Also, it works only if $x, y$ belongs to same domain.
The below one is a tautology provided $x,y$ have the same domain.
$\left(\forall x \,\forall y P(x,y) \, \right) \rightarrow \,\left( \forall x \forall y \,P(y, x)\right)$
If P(x,y) is true for all (x,y), then P(y,x) is true for all (x,y).