\begin{definition}[Definition:Conditional/Truth Table/Matrix Form]
The characteristic truth table of the conditional (implication) operator $p \implies q$ can be presented in matrix form as follows:
:$\begin{array}{c|cc}
\implies & \T & \F \\
\hline
\T & \T & \F \\
\F & \T & \T \\
\end{array}$
As $\implies$ is not commutative, it is also instructive to give a truth table 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 $p \impliedby q$ can be presented in matrix form as follows:
:$\begin{array}{c|cc}
\impliedby & \T & \F \\
\hline
\T & \T & \T \\
\F & \F & \T \\
\end{array}
\qquad
\begin{array}{c|cc}
\neg \implies & \T & \F \\
\hline
\T & \F & \T \\
\F & \F & \F \\
\end{array}
\qquad
\begin{array}{c|cc}
\neg \impliedby & \T & \F \\
\hline
\T & \F & \F \\
\F & \T & \F \\
\end{array}$
\end{definition}