Discrete mathematics logic

Question

What is the negation for the question P <-> ~Q in logic. Please give some suggestions how to solve this.... In logic how to find negation of the terms.

Answer 1

I would do $$(P\iff \neg Q) \equiv [(P\to \neg Q) \land (\neg Q\to P)].$$ Hence, negating looks like this: \begin{align*} \neg(P\iff \neg Q) &\equiv \neg[(P\to \neg Q) \land (\neg Q\to P)] \\ &\equiv \neg(P\to \neg Q) \lor \neg(\neg Q\to P) \\ &\equiv (P\land \neg\neg Q) \lor (\neg Q \land \neg P) \\ &\equiv (P\land Q) \lor (\neg Q\land \neg P). \end{align*} I think you'll find this is also equivalent to $\neg(P\otimes Q),$ where $\otimes$ is the exclusive OR. Or even simpler: $P\iff Q.$