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# Truth table

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:

 $$p\!$$ $$\lnot p\!$$ $$\operatorname{F}$$ $$\operatorname{T}$$ $$\operatorname{T}$$ $$\operatorname{F}$$

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:

 $$\text{Notation}\!$$ $$\text{Vocalization}\!$$ $$\bar{p}\!$$ $$p\!$$ 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:

 $$p\!$$ $$q\!$$ $$p \land q$$ $$\operatorname{F}$$ $$\operatorname{F}$$ $$\operatorname{F}$$ $$\operatorname{F}$$ $$\operatorname{T}$$ $$\operatorname{F}$$ $$\operatorname{T}$$ $$\operatorname{F}$$ $$\operatorname{F}$$ $$\operatorname{T}$$ $$\operatorname{T}$$ $$\operatorname{T}$$

## 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:

 $$p\!$$ $$q\!$$ $$p \lor q$$ $$\operatorname{F}$$ $$\operatorname{F}$$ $$\operatorname{F}$$ $$\operatorname{F}$$ $$\operatorname{T}$$ $$\operatorname{T}$$ $$\operatorname{T}$$ $$\operatorname{F}$$ $$\operatorname{T}$$ $$\operatorname{T}$$ $$\operatorname{T}$$ $$\operatorname{T}$$

## 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:

 $$p\!$$ $$q\!$$ $$p = q\!$$ $$\operatorname{F}$$ $$\operatorname{F}$$ $$\operatorname{T}$$ $$\operatorname{F}$$ $$\operatorname{T}$$ $$\operatorname{F}$$ $$\operatorname{T}$$ $$\operatorname{F}$$ $$\operatorname{F}$$ $$\operatorname{T}$$ $$\operatorname{T}$$ $$\operatorname{T}$$

## 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 $$p \ne q,\!$$ appears below:

 $$p\!$$ $$q\!$$ $$p ~\operatorname{XOR}~ q$$ $$\operatorname{F}$$ $$\operatorname{F}$$ $$\operatorname{F}$$ $$\operatorname{F}$$ $$\operatorname{T}$$ $$\operatorname{T}$$ $$\operatorname{T}$$ $$\operatorname{F}$$ $$\operatorname{T}$$ $$\operatorname{T}$$ $$\operatorname{T}$$ $$\operatorname{F}$$

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:

 $$p\!$$ $$q\!$$ $$p \Rightarrow q\!$$ $$\operatorname{F}$$ $$\operatorname{F}$$ $$\operatorname{T}$$ $$\operatorname{F}$$ $$\operatorname{T}$$ $$\operatorname{T}$$ $$\operatorname{T}$$ $$\operatorname{F}$$ $$\operatorname{F}$$ $$\operatorname{T}$$ $$\operatorname{T}$$ $$\operatorname{T}$$

## 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:

 $$p\!$$ $$q\!$$ $$p \stackrel{\circ}{\curlywedge} q\!$$ $$\operatorname{F}$$ $$\operatorname{F}$$ $$\operatorname{T}$$ $$\operatorname{F}$$ $$\operatorname{T}$$ $$\operatorname{T}$$ $$\operatorname{T}$$ $$\operatorname{F}$$ $$\operatorname{T}$$ $$\operatorname{T}$$ $$\operatorname{T}$$ $$\operatorname{F}$$

## 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:

 $$p\!$$ $$q\!$$ $$p \curlywedge q\!$$ $$\operatorname{F}$$ $$\operatorname{F}$$ $$\operatorname{T}$$ $$\operatorname{F}$$ $$\operatorname{T}$$ $$\operatorname{F}$$ $$\operatorname{T}$$ $$\operatorname{F}$$ $$\operatorname{F}$$ $$\operatorname{T}$$ $$\operatorname{T}$$ $$\operatorname{F}$$

## Document history

Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.