# Truth table

☞ This page belongs to resource collections on Logic and Inquiry.
A truth table is a tabular array that illustrates the computation of a logical function, that is, a function of the form $f:\mathbb {A} ^{k}\to \mathbb {A} ,$ where $k\!$ is a non-negative integer and $\mathbb {A}$ is the domain of logical values $\{\operatorname {false} ,\operatorname {true} \}.$ The names of the logical values, or truth values, are commonly abbreviated in accord with the equations $\operatorname {F} =\operatorname {false}$ and $\operatorname {T} =\operatorname {true} .$
In many applications it is usual to represent a truth function by a boolean function, that is, a function of the form $f:\mathbb {B} ^{k}\to \mathbb {B} ,$ where $k\!$ is a non-negative integer and $\mathbb {B}$ is the boolean domain $\{0,1\}.\!$ In most applications $\operatorname {false}$ is represented by $0\!$ and $\operatorname {true}$ is represented by $1\!$ but the opposite representation is also possible, depending on the overall representation of truth functions as boolean functions. The remainder of this article assumes the usual representation, taking the equations $\operatorname {F} =0$ and $\operatorname {T} =1$ for granted.

## Logical negation

Logical negation is an operation on one logical value, typically the value of a proposition, that produces a value of true when its operand is false and a value of false when its operand is true.
The truth table of $\operatorname {NOT} ~p,$ also written $\lnot p,\!$ appears below:
${\text{Logical Negation}}\!$ :
| $p\!$ | $\lnot p\!$ |
| $\operatorname {F}$ | $\operatorname {T}$ |
| $\operatorname {T}$ | $\operatorname {F}$ |
The negation of a proposition $p\!$ may be found notated in various ways in various contexts of application, often merely for typographical convenience. Among these variants are the following:
${\text{Variant Notations}}\!$ :
| ${\text{Notation}}\!$ | ${\text{Vocalization}}\!$ |
| ${\bar {p}}\!$ | $p\!$ bar |
| ${\tilde {p}}\!$ | $p\!$ tilde |
| $p'\!$ | $p\!$ prime $p\!$ complement |
| $!p\!$ | bang $p\!$ |

## Logical conjunction

Logical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are true.
The truth table of $p~\operatorname {AND} ~q,$ also written $p\land q\!$ or $p\cdot q,\!$ appears below:
${\text{Logical Conjunction}}\!$ :
| $p\!$ | $q\!$ | $p\land q$ |
| $\operatorname {F}$ | $\operatorname {F}$ | $\operatorname {F}$ |
| $\operatorname {F}$ | $\operatorname {T}$ | $\operatorname {F}$ |
| $\operatorname {T}$ | $\operatorname {F}$ | $\operatorname {F}$ |
| $\operatorname {T}$ | $\operatorname {T}$ | $\operatorname {T}$ |

## Logical disjunction

Logical disjunction, also called logical alternation, is an operation on two logical values, typically the values of two propositions, that produces a value of false if and only if both of its operands are false.
The truth table of $p~\operatorname {OR} ~q,$ also written $p\lor q,\!$ appears below:
${\text{Logical Disjunction}}\!$ :
| $p\!$ | $q\!$ | $p\lor q$ |
| $\operatorname {F}$ | $\operatorname {F}$ | $\operatorname {F}$ |
| $\operatorname {F}$ | $\operatorname {T}$ | $\operatorname {T}$ |
| $\operatorname {T}$ | $\operatorname {F}$ | $\operatorname {T}$ |
| $\operatorname {T}$ | $\operatorname {T}$ | $\operatorname {T}$ |

## Logical equality

Logical equality is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false or both operands are true.
The truth table of $p~\operatorname {EQ} ~q,$ also written $p=q,\!$ $p\Leftrightarrow q,\!$ or $p\equiv q,\!$ appears below:
${\text{Logical Equality}}\!$ :
| $p\!$ | $q\!$ | $p=q\!$ |
| $\operatorname {F}$ | $\operatorname {F}$ | $\operatorname {T}$ |
| $\operatorname {F}$ | $\operatorname {T}$ | $\operatorname {F}$ |
| $\operatorname {T}$ | $\operatorname {F}$ | $\operatorname {F}$ |
| $\operatorname {T}$ | $\operatorname {T}$ | $\operatorname {T}$ |

## Exclusive disjunction

Exclusive disjunction, also known as logical inequality or symmetric difference, is an operation on two logical values, typically the values of two propositions, that produces a value of true just in case exactly one of its operands is true.
The truth table of $p~\operatorname {XOR} ~q,$ also written $p+q\!$ or $p\neq q,\!$ appears below:
${\text{Exclusive Disjunction}}\!$ :
| $p\!$ | $q\!$ | $p~\operatorname {XOR} ~q$ |
| $\operatorname {F}$ | $\operatorname {F}$ | $\operatorname {F}$ |
| $\operatorname {F}$ | $\operatorname {T}$ | $\operatorname {T}$ |
| $\operatorname {T}$ | $\operatorname {F}$ | $\operatorname {T}$ |
| $\operatorname {T}$ | $\operatorname {T}$ | $\operatorname {F}$ |
The following equivalents may then be deduced:
* ${\begin{matrix}p+q&=&(p\land \lnot q)&\lor &(\lnot p\land q)\\[6pt]&=&(p\lor q)&\land &(\lnot p\lor \lnot q)\\[6pt]&=&(p\lor q)&\land &\lnot (p\land q)\end{matrix}}$

## Logical implication

The logical implication relation and the material conditional function are both associated with an operation on two logical values, typically the values of two propositions, that produces a value of false if and only if the first operand is true and the second operand is false.
The truth table associated with the material conditional ${\text{if}}~p~{\text{then}}~q,\!$ symbolized $p\rightarrow q,\!$ and the logical implication $p~{\text{implies}}~q,\!$ symbolized $p\Rightarrow q,\!$ appears below:
${\text{Logical Implication}}\!$ :
| $p\!$ | $q\!$ | $p\Rightarrow q\!$ |
| $\operatorname {F}$ | $\operatorname {F}$ | $\operatorname {T}$ |
| $\operatorname {F}$ | $\operatorname {T}$ | $\operatorname {T}$ |
| $\operatorname {T}$ | $\operatorname {F}$ | $\operatorname {F}$ |
| $\operatorname {T}$ | $\operatorname {T}$ | $\operatorname {T}$ |

## Logical NAND

The logical NAND is an operation on two logical values, typically the values of two propositions, that produces a value of false if and only if both of its operands are true. In other words, it produces a value of true if and only if at least one of its operands is false.
The truth table of $p~\operatorname {NAND} ~q,$ also written $p{\stackrel {\circ }{\curlywedge }}q\!$ or $p\barwedge q,\!$ appears below:
${\text{Logical NAND}}\!$ :
| $p\!$ | $q\!$ | $p{\stackrel {\circ }{\curlywedge }}q\!$ |
| $\operatorname {F}$ | $\operatorname {F}$ | $\operatorname {T}$ |
| $\operatorname {F}$ | $\operatorname {T}$ | $\operatorname {T}$ |
| $\operatorname {T}$ | $\operatorname {F}$ | $\operatorname {T}$ |
| $\operatorname {T}$ | $\operatorname {T}$ | $\operatorname {F}$ |

## Logical NNOR

The logical NNOR (“Neither Nor”) is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are false. In other words, it produces a value of false if and only if at least one of its operands is true.
The truth table of $p~\operatorname {NNOR} ~q,$ also written $p\curlywedge q,\!$ appears below:
${\text{Logical NNOR}}\!$ :
| $p\!$ | $q\!$ | $p\curlywedge q\!$ |
| $\operatorname {F}$ | $\operatorname {F}$ | $\operatorname {T}$ |
| $\operatorname {F}$ | $\operatorname {T}$ | $\operatorname {F}$ |
| $\operatorname {T}$ | $\operatorname {F}$ | $\operatorname {F}$ |
| $\operatorname {T}$ | $\operatorname {T}$ | $\operatorname {F}$ |

## Translations

* 中文 : 真值表

## Syllabus
### Focal nodes

* Inquiry Live
* Logic Live

### Peer nodes

* Truth Table @ InterSciWiki
* Truth Table @ Subject Wikis
* Truth Table @ Wikiversity
* Truth Table @ Wikiversity Beta