Discussion
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?
Answer: A
The expression is \((\exists y)(\forall x)((x\neq y)\rightarrow(M(x,y)\wedge\neg M(y,x)))\).
Read the statement as: there is a person y such that, for every other person x, x knows y and y does not know x. 'There is a person' is an existential quantifier on y, placed outside; 'everyone else' is a universal quantifier on x inside it; and 'else' becomes the condition x ≠ y. M(x, y) says x knows y, and ¬M(y, x) says y does not know x.
Swapping to ∀y ∃x says every person has someone who knows them and whom they do not know, a much weaker claim about everybody. ∃y ∃x only needs one such pair. ∀y ∀x would require every person to be known by, and not know, every other person, which is impossible for two people.
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