Definition:Conditional/Truth Table/Matrix Form

Definition
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}$