Truth table

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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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

中文 : 真值表

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