# truth table

A 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:

$a$ $b$ $a\lor b$
F F F
F T T
T F T
T T T

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

$a$ $b$ $c$ $(a\land b)\rightarrow c$
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

(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:

$p$ $q$ $\neg p$ $(p\land q)$ $\neg p\lor(p\land q)$
F F T F T
F T T F T
T F F F F
T T F T T

For reference, here is a truth table of some popular connectives:

$p$ $q$ $p\lor q$ $p\land q$ $p\veebar q$ $p\rightarrow q$ $p\leftrightarrow q$
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

For completeness, here are the remaining connectives, excluding trivial connectives which depend on only one or none of their arguments:

$p$ $q$ $p\not\!\!\land q$ $p\not\!\lor q$ $p\leftarrow q$ $p\not\rightarrow q$ $p\not\!\leftarrow q$
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
Title truth table TruthTable 2013-03-22 11:54:35 2013-03-22 11:54:35 rspuzio (6075) rspuzio (6075) 16 rspuzio (6075) Definition msc 03-00 msc 34C29 ZerothOrderLogic PropositionalCalculus