Am I correct in the following:
If the domain is all students, and C(x,y) is the predicate of x having chatted with y. Then the sentence
There is a student in this class who has chatted with exactly one other student
Can be represented as
$$\exists x\exists y(y\ne x\land \forall z(z\ne x\to(z=y\leftrightarrow C(x,z))))$$
Which is logically equivalent to:
$$\exists x \exists y[(x\neq y) \land\forall z ([z=x] \lor[(z\neq y) \lor C(x,y)] \land [\lnot C(x,y) \lor (z = y)])]$$
The negation of which would be:
$$\forall x \forall y [(x=y) \lor \exists z([z\neq x] \land [(z =y) \land \lnot C(x,y)] \lor[C(x,y) \land(z \neq y)])]$$
That is also logically equivalent to:
$$\forall x \forall y [(x = y) \lor \exists z([z \neq x] \land [C(x,y) \leftrightarrow (z \neq y)])]$$
Which can be translated to: All students have spoken with at least one other student or themselves?
Best Answer
The negation would be: For all students x, either x has chatted with no other students or x has chatted with more than one other student.
[This is just ruling out x chatting with exactly one other student.]