Definition:Conditional/Truth Table

Definition
As $\implies$ is not commutative, it is instructive to give a truth table for both $p \implies q$ and $p \impliedby q$ (which of course is the same as $q \implies p$).

The truth tables of the conditional (implication) operator $p \implies q$ and $p \impliedby q$ and their complements is as follows:


 * $\begin{array}{|cc||c|c||c|c|} \hline

p & q & p \implies q & \neg \left({p \implies q}\right) & p \impliedby q & \neg \left({p \impliedby q}\right) \\ \hline F & F & T & F & T & F \\ F & T & T & F & F & T \\ T & F & F & T & T & F \\ T & T & T & F & T & F \\ \hline \end{array}$