Definition:Conditional/Truth Table

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

The truth table of the conditional (implication) operator $p \implies q$ and $p \ \Longleftarrow \ q$ 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 \ \Longleftarrow \ q & \neg \left({p \ \Longleftarrow \ 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}$

Matrix Form

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

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