For two different persons 𝑥 and 𝑦, the predicate 𝑀(𝑥, 𝑦) denotes that x…

Step-by-Step Solution

Let's break down the statement: "There is a person who does not know anyone else, but that person is known by everyone else."

1. Analyze the Subject

The phrase "There is a person" indicates an existential quantifier (∃). Let's call this person y. So, we start with (∃y).

2. Analyze the Condition for Others

The phrase "everyone else" refers to all other persons x. This requires a universal quantifier (∀). So, for the person y, the condition must hold for all x where x ≠ y.

3. Translate the Relationships

The statement has two parts for the relationship between x and y (where x ≠ y):

• "that person is known by everyone else": This means x knows y, represented as M(x, y).

• "who does not know anyone else": This means y does not know x, represented as ¬M(y, x).

These two conditions must be true simultaneously, so we use the AND operator (∧).

4. Construct the Final Expression

Combining the quantifiers and the condition:

For all x (where x ≠ y), M(x, y) is true AND ¬M(y, x) is true.

There exists a y such that for all x (where x ≠ y), the condition holds.

Expression: (∃y)(∀x) ((x ≠ y) → (M(x,y) ∧ ¬M(y,x)))

Conclusion

The correct expression corresponds to Option A.