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: Translate the English statement into predicate logic piece by piece. The statement says: there is a person who does not know anyone else, but that person is known by everyone else.
Step 2: Let this special person be denoted by y. For every other person x (with x not equal to y), two things must hold: x knows y, that is \(M(x,y)\), and y does not know x, that is \(\neg M(y,x)\).
Step 3: Since this condition must hold for every other person x, we attach a universal quantifier \(\forall x\) to the implication \((x \neq y) \to (M(x,y) \land \neg M(y,x))\). Since we only need one such special person to exist, we attach an existential quantifier \(\exists y\) on the outside.
Step 4: Putting it together gives \[(\exists y)(\forall x)\big((x \neq y) \to (M(x,y) \land \neg M(y,x))\big)\] which is exactly option (A). The other options either swap the quantifiers (making every person the special one, or asking for existence of x too) and do not capture the intended meaning.
Final Answer: Option (A): \((\exists y)(\forall x)((x \neq y) \to (M(x,y) \land \neg M(y,x)))\)