What's the proper way to prove P$\rightarrow$Q is not tautologically equivalent to Q$\rightarrow$P I want to rigorously show that [$(p \rightarrow q)\equiv(q \rightarrow q)$] is not true. $Let A =(p \rightarrow q), B =(q \rightarrow q)$ I tried to prove this by supposing $A$ and showing that $B$ does not follow. [$(p \rightarrow q) \rightarrow (q \rightarrow p)$] I expanded that but it didn't $= 0$ I was told to look at Reductio Ad Absurdum, but I don't see how that applies in this case. This is a real-world example. This is not homework.

The most straightforward way is to construct the truth tables and see that they are not equal. Or you could show values of $P$ and $Q$ that make them different (which would amount to mention the specific line of the truth tables where they differ).

If $P$ is true and $Q$ is false, then $P\to Q$ is false, while $Q\to P$ is true.

Edit: Changed $Q\to Q$ to $Q\to P$, to go with the title.