How to prove implication using Proof by Cases

Question

Assuming that $P \land \neg Q \leftrightarrow 0$, how would we go about proving $P \leftrightarrow Q$ using proof-by-cases?

Answer 1

Complete the table and compare the last two columns.

$$\begin{array}{l:l} P&Q&P\wedge\neg Q& P\wedge\neg Q\leftrightarrow 0 & P\leftrightarrow Q\\ \hline 1 & 1 & 0 & 1 & 1\\ 1 & 0 & & & \\ 0 & 1 & & & \\ 0 & 0 & & & \end{array}$$

Now use the results to guide you to the cases.