# Definition:Conditional/Truth Table

## Definition

The characteristic truth table of the conditional (implication) operator $p \implies q$ is as follows:

$\begin{array}{|cc||c|} \hline p & q & p \implies q \\ \hline \F & \F & \T \\ \F & \T & \T \\ \T & \F & \F \\ \T & \T & \T \\ \hline \end{array}$

As $\implies$ is not commutative, it is also instructive to give a characteristic truth tables 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 $p \impliedby q$ are as follows:

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

### Matrix Form

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

### Truth Table Number

The truth table number of the conjunction operator $p \land q$ is as follows:

Ascending order:

$1101$ or $\T \T \F \T$

Descending order:

$1011$ or $\T \F \T \T$