\begin{definition}[Definition:Conditional/Truth Table]
The characteristic truth table of the conditional (implication) operator $p \implies q$ is as follows:
:$\begin{array}{|cc||c|} \hline
p & q & p \implies q \\
\hline
\F & \F & \T \\
\F & \T & \T \\
\T & \F & \F \\
\T & \T & \T \\
\hline
\end{array}$
As $\implies$ is not commutative, it is also instructive to give a characteristic truth tables for $p \impliedby q$ (which of course is the same as $q \implies p$).
Hence the characteristic truth tables of the conditional (implication) operator $p \impliedby q$ and the complements of both $p \implies q$ and $p \impliedby q$ are as follows:
:$\begin{array}{|cc||c||c|c|} \hline
p & q & \neg \paren {p \implies q} & p \impliedby q & \neg \paren {p \impliedby q} \\
\hline
\F & \F & \F & \T & \F \\
\F & \T & \F & \F & \T \\
\T & \F & \T & \T & \F  \\
\T & \T & \F & \T & \F \\
\hline
\end{array}$
\end{definition}