Discrete Math: Implication If $\neg(P) \to \neg(Q) = Q \to P$ works as a Rule, then why doesn't $\neg(P) \to \neg(Q) = P \to Q$ work as a rule.
 A: You should understand this 
Implication is equal to Contra positive 
Implication is not equal to converse and inverse.
You can prove this with truth table. 
Logical  proving,
 $$\begin{align}
& P \;\to\; Q
\\[1ex] 
  = \quad& \neg P \vee Q & \text{Implication Equivalence}
\\[1ex]
  = \quad& \neg \neg Q \vee \neg P & \text{Commutivity and Double Negation}
\\[1ex]
  = \quad& \neg  Q \;\to\; \neg P & \text{Implication Equivalence}
\end{align}$$
Which proves that  $P\to Q \;=\; \neg Q \to \neg P$
A: This is best understood in terms of the truth tables for each of these statements.  Let's try the statement $(\neg P \Rightarrow \neg Q) \Leftrightarrow (P \Rightarrow Q)$.
\begin{array}{|c|c|c|c|c|c|c|}
\hline
P & Q & \neg P & \neg Q & \neg P\Rightarrow \neg Q & P \Rightarrow Q & \Leftrightarrow \\ \hline
T & T & F & F & T & T & T \\ \hline
T & F & F & T & T & F & F \\ \hline
F & T & T & F & F & T & F \\ \hline
F & F & T & T & T & T & T \\ \hline
\end{array}
You can see that these statement are not logically equivalent.  Can you prove that $(\neg P \Rightarrow \neg Q) \Leftrightarrow (Q \Rightarrow P)$?
