| Version 7 |
Version 6 |
| 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. |
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. |
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| For instance, the truth table of the connective ``or'' is as follows: |
For instance, the truth table of the connective ``or'' is as follows: |
| \begin{center} |
\begin{center} |
| \begin{tabular}{ccc} |
\begin{tabular}{ccc} |
| $a$ & $b$ & $a \lor b$ \\ |
$a$ & $b$ & $a \lor b$ \\ |
| \hline |
\hline |
| F & F & F \\ |
F & F & F \\ |
| F & T & T \\ |
F & T & T \\ |
| T & F & T \\ |
T & F & T \\ |
| T & T & T |
T & T & T |
| \end{tabular} |
\end{tabular} |
| \end{center} |
\end{center} |
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| For $n$ input variables, there will always be $2^n$ rows in the truth table. |
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 |
A sample truth table for ``$(a \land b) \rightarrow c$'' would be |
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| \begin{center} |
\begin{center} |
| \begin{tabular}{cccc} |
\begin{tabular}{cccc} |
| $a$ & $b$ & $c$ & $(a \land b) \rightarrow c$ \\ |
$a$ & $b$ & $c$ & $(a \land b) \rightarrow c$ \\ |
| \hline |
\hline |
| F & F & F & T \\ |
F & F & F & T \\ |
| F & F & T & F \\ |
F & F & T & F \\ |
| F & T & F & T \\ |
F & T & F & T \\ |
| F & T & T & F \\ |
F & T & T & F \\ |
| T & F & F & T \\ |
T & F & F & T \\ |
| T & F & T & F \\ |
T & F & T & F \\ |
| T & T & F & T \\ |
T & T & F & T \\ |
| T & T & T & T |
T & T & T & T |
| \end{tabular} |
\end{tabular} |
| \end{center} |
\end{center} |
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| (Note that $\land$ represents logical and, while $\rightarrow$ represents the conditional truth function). |
(Note that $\land$ represents logical and, while $\rightarrow$ represents the conditional truth function). |
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| To compute truth tables of expressions, one often proceeds in steps. for instance, |
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| to compute a truth table for ``$\neg p \lor (p \land q)$, one might proceed as follows: |
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| \begin{center} |
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| \begin{tabular}{ccccc} |
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| $p$ & $q$ & $\neg p$ & $(p \land q)$ & $\neg p \lor (p \land q)$ \\ |
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| \hline |
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| F & F & T & F & T \\ |
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| F & T & T & F & T \\ |
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| T & F & F & F & F \\ |
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| T & T & F & T & T |
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| \end{tabular} |
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| \end{center} |
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