Question

For two different persons 𝑥 and 𝑦, the predicate 𝑀(𝑥, 𝑦) denotes that x knows y.
Consider the following statement.
There is a person who does not know anyone else, but that person is known by
everyone else.
Which one of the following expressions represents the above statement?

• A. (∃𝑦)(∀𝑥) ((𝑥≠𝑦) →(𝑀(𝑥, 𝑦) ∧¬𝑀(𝑦, 𝑥)))
• B. (∀𝑦)(∃𝑥) ((𝑥≠𝑦) →(𝑀(𝑥, 𝑦) ∧¬𝑀(𝑦, 𝑥)))
• C. (∃𝑦)(∃𝑥) ((𝑥≠𝑦) →(𝑀(𝑥, 𝑦) ∧¬𝑀(𝑦, 𝑥)))
• D. (∀𝑦)(∀𝑥) ((𝑥≠𝑦) →(𝑀(𝑥, 𝑦) ∧¬𝑀(𝑦, 𝑥)))

Hint:

Fix the special person as y. For every other person x, x must know y (M(x,y)) while y must not know x (not M(y,x)); this needs an outer 'there exists y' and an inner 'for all x'.

Answer 1

The Correct Option is A

Step 1: Break the sentence into two separate claims about the special person, call them y. Claim 1: y does not know anyone else, meaning for every other person x, y does not know x, written \(\neg M(y,x)\). Claim 2: y is known by everyone else, meaning for every other person x, x knows y, written \(M(x,y)\).
Step 2: Both claims apply to the same set of other people x, so they combine with AND inside a universally quantified implication over x: for all x with \(x \neq y\), \(M(x,y) \land \neg M(y,x)\) must be true.
Step 3: The word 'there is a person' tells us only one such y needs to exist, not that this holds for every y, so the outer quantifier on y must be existential, not universal. This immediately rules out any option starting with \((\forall y)\).
Step 4: Among the remaining existential-y options, only the one using \((\forall x)\) correctly forces the condition on every other person; using \((\exists x)\) would only require one other person to satisfy it, which is too weak.
Final Answer: Option (A): \((\exists y)(\forall x)((x \neq y) \to (M(x,y) \land \neg M(y,x)))\)