truth tableTruthTable2001-10-26 06:52:442008-03-29 12:26:30Definition\usepackage{amssymb}
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\usepackage{xypic}A \emph{truth table} is a tabular listing of all possible input value combinations for a logical function and their corresponding output values. Similarly, the truth table of a logical proposition is the truth table of the corresponding logical function.
For instance, the truth table of the connective ``or'' is as follows:
\begin{center}
\begin{tabular}{ccc}
$a$ & $b$ & $a \lor b$ \\
\hline
F & F & F \\
F & T & T \\
T & F & T \\
T & T & T
\end{tabular}
\end{center}
For $n$ input variables, there will always be $2^n$ rows in the truth table.
A sample truth table for ``$(a \land b) \rightarrow c$'' would be
\begin{center}
\begin{tabular}{cccc}
$a$ & $b$ & $c$ & $(a \land b) \rightarrow c$ \\
\hline
F & F & F & T \\
F & F & T & F \\
F & T & F & T \\
F & T & T & F \\
T & F & F & T \\
T & F & T & F \\
T & T & F & T \\
T & T & T & T
\end{tabular}
\end{center}
(Note that $\land$ represents logical and, while $\rightarrow$ represents the conditional truth function).
To compute truth tables of expressions, one often proceeds in steps. for instance,
to compute a truth table for ``$\neg p \lor (p \land q)$, one might proceed as follows:
\begin{center}
\begin{tabular}{ccccc}
$p$ & $q$ & $\neg p$ & $(p \land q)$ & $\neg p \lor (p \land q)$ \\
\hline
F & F & T & F & T \\
F & T & T & F & T \\
T & F & F & F & F \\
T & T & F & T & T
\end{tabular}
\end{center}
For reference, here is a truth table of some popular connectives:
\begin{center}
\begin{tabular}{ccccccc}
$p$ & $q$ & $p \lor q$ & $p \land q$ & $p \veebar q$ & $p \rightarrow q$ & $p \leftrightarrow q$ \\
\hline
F & F & F & F & F & T & T \\
F & T & T & F & T & T & F \\
T & F & T & F & T & F & F \\
T & T & T & T & F & T & T
\end{tabular}
\end{center}
For completeness, here are the remaining connectives, excluding trivial connectives which
depend on only one or none of their arguments:
\begin{center}
\begin{tabular}{ccccccccc}
$p$ & $q$ & $p \not\!\!\land q$ & $p \not\!\lor q$ & $p \leftarrow q$ & $p \not\rightarrow q$ & $p \not\!\leftarrow q$ \\
\hline
F & F & T & T & T & F & F \\
F & T & T & F & F & F & T \\
T & F & T & F & T & T & F \\
T & T & F & F & T & F & F
\end{tabular}
\end{center}