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. \((\exists y)(\forall x)\, ((x \neq y) \rightarrow (M(x, y) \wedge \neg M(y, x)))\)
• B. \((\forall y)(\exists x)\, ((x \neq y) \rightarrow (M(x, y) \wedge \neg M(y, x)))\)
• C. \((\exists y)(\exists x)\, ((x \neq y) \rightarrow (M(x, y) \wedge \neg M(y, x)))\)
• D. \((\forall y)(\forall x)\, ((x \neq y) \rightarrow (M(x, y) \wedge \neg M(y, x)))\)

Hint:

The sentence describes exactly one special person, so use one existential quantifier for that person and one universal quantifier ranging over everyone else.

Answer 1

The Correct Option is A

Step 1: Identify the two parts of the English statement.
The statement has two clauses joined by "but": (1) "There is a person who does not know anyone else" and (2) "that same person is known by everyone else." Both clauses talk about ONE specific person, so the outer quantifier over that person must be an existential \(\exists\), not a universal \(\forall\), since we are asserting only that such a person exists, not that every person has this property.

Step 2: Fix the outer variable.
Call this special person \(y\), matching the variable used in the options. So the statement begins \((\exists y)\), meaning "there exists a person \(y\) such that...". This already rules out options (B) and (D), since they put \(\forall y\) as the outer quantifier, which would incorrectly claim the property holds for every person, not just one special person.

Step 3: Translate "does not know anyone else."
For every other person \(x\) (that is, every \(x\) with \(x \neq y\)), \(y\) does not know \(x\), written \(\neg M(y, x)\). Since this must hold for every such \(x\), the quantifier on \(x\) must be universal, \((\forall x)\), not existential. This rules out option (C), which uses \((\exists x)\) and so would only require the property to hold for at least one other person, not all of them.

Step 4: Translate "is known by everyone else."
For every other person \(x\), \(x\) knows \(y\), written \(M(x, y)\). Combined with Step 3, for every \(x \neq y\) we need both \(M(x, y)\) (x knows y) and \(\neg M(y, x)\) (y does not know x) to hold together, joined by \(\wedge\).

Step 5: Assemble the full formula.
Putting the quantifiers and the conjunction together: \[ (\exists y)(\forall x)\, ((x \neq y) \rightarrow (M(x, y) \wedge \neg M(y, x))) \] This says: there is a person \(y\) such that for every other person \(x\), \(x\) knows \(y\) and \(y\) does not know \(x\), exactly matching the English statement. This is option (A).

Step 6: Why the other options fail.
Option (B), \((\forall y)(\exists x)\), would claim that for every person \(y\), there merely exists some other person satisfying the knows/does-not-know pattern with \(y\), a much stronger and different claim about everyone, not the single special person described. Option (C), \((\exists y)(\exists x)\), only needs one other witness \(x\) to satisfy the condition, so it would be true even if \(y\) knew and was known by every other person except one, which does not match "does not know anyone else" and "known by everyone else." Option (D), \((\forall y)(\forall x)\), asserts this knowledge pattern holds between every pair of distinct people simultaneously, which is far stronger than asserting a single such person exists.

Final Answer:
The correct translation uses one existential outer quantifier and one universal inner quantifier.
\[ \boxed{(\exists y)(\forall x)\, ((x \neq y) \rightarrow (M(x, y) \wedge \neg M(y, x)))} \]