I have to answer this question.
Let the domain of the following predicates be defined as P, the set of married people:
everyone in P is married to someone else in P. For any two people
a ∈ P & b ∈ P.
let M(a,b) denote that a is married (to b). You should note that a is not the same as b.
M(a,b)=( true if a is married to b, false otherwise)
Question:
Everyone in P is married to someone else in P.
The correct answer was this:
∀ y. ∃ x. M(y,x)
The answer I wrote was this:
$∀ x. ∀ y. x \neq y \implies M(x,y)$
To me the correct answer reads like this:
For all y, there exists an x where y is married to x.
The way I read my answer is:
For all x, for all y if x doesn't equal y then x is married to y.
Is my answer correct?
Your answer $$\forall x\forall y\Big((x\neq y)\to M(x, y)\Big)$$
says, essentially "Everyone is married to everyone, except to him/herself."
Yes, you should change $\forall y$ to $\exists y$, and you should should use "$\land$", and not "$\to$."
Then you can correctly express the proposition:
$$\forall x, \exists y\Big((x\neq y) \land M(x, y)\Big)$$