# Definition:Conditional/Truth Table/Matrix Form

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## 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:

$\begin{array}{c|cc} \impliedby & \T & \F \\ \hline \T & \T & \T \\ \F & \F & \T \\ \end{array} \qquad \begin{array}{c|cc} \neg \implies & \T & \F \\ \hline \T & \F & \T \\ \F & \F & \F \\ \end{array} \qquad \begin{array}{c|cc} \neg \impliedby & \T & \F \\ \hline \T & \F & \F \\ \F & \T & \F \\ \end{array}$