Correct : a

The English statement has two parts about the same special person y — they do not know anyone else, but everyone else knows them.
The first word "there is" tells us we need an existential quantifier for y, so we start with ∃y. The phrases "anyone else" and "everyone else" both refer to all persons x different from y, so we need ∀x with the condition x ≠ y. Using implication (x ≠ y) → ... is the standard way to restrict the universal quantifier to only those x that are different from y.
Inside the implication we need two things to hold simultaneously for every such x — M(x, y) meaning x knows y (everyone else knows that person), and ¬M(y, x) meaning y does not know x (that person does not know anyone else). Both conditions are joined with ∧.
So the full expression is (∃y)(∀x)((x ≠ y) → (M(x, y) ∧ ¬M(y, x))), which is option A.
Why the others are wrong:
Option B uses ∀y∃x which says "for every person y, there exists some x" — this changes the meaning completely and does not fix one special person.
Option C uses ∃y∃x which says there exist some y and some x — this only claims two specific people have this relationship, not that it holds for all others.
Option D uses ∀y∀x which universalises over everyone — this would mean every person simultaneously does not know anyone and is known by everyone, which is far too strong and not what is stated.
Correct answer: A ✓