# Truth table

This document is part of a corpus of sample documents. These documents can be used for evaluating mathematical knowledge management tools.
Note that not every document in this corpus needs to be mathematically sound. While this would, e.g., make sense for testing automated theorem provers, it is not always necessary for testing document management tools.

A truth table is a tabular array that illustrates the computation of 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 $\{ 0, 1 \}.\!$

## 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 NOT p (also written as ~p or ¬p) is as follows:

Logical Negation
p ¬p
F T
T F

The logical negation of a proposition p is notated in different ways in various contexts of discussion and fields of application. Among these variants are the following:

Variant Notations
Notation Vocalization
$\bar{p}$ bar p
$p'\!$ 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 AND q (also written as p ∧ q, p & q, or p$\cdot$q) is as follows:

Logical Conjunction
p q p ∧ q
F F F
F T F
T F F
T T 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 OR q (also written as p ∨ q) is as follows:

Logical Disjunction
p q p ∨ q
F F F
F T T
T F T
T T 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 EQ q (also written as p = q, p ↔ q, or p ≡ q) is as follows:

Logical Equality
p q p = q
F F T
F T F
T F F
T T 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 XOR q (also written as p + q, p ⊕ q, or p ≠ q) is as follows:

Exclusive Disjunction
p q p XOR q
F F F
F T T
T F T
T T F

The following equivalents can then be deduced:

$\begin{matrix} p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\ \\ & = & (p \lor q) & \land & (\lnot p \lor \lnot q) \\ \\ & = & (p \lor q) & \land & \lnot (p \land q) \end{matrix}$

## Logical implication

The logical implication and the material conditional 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 if p then q (symbolized as p → q) and the logical implication p implies q (symbolized as p ⇒ q) is as follows:

Logical Implication
p q p ⇒ q
F F T
F T T
T F F
T T T

## Logical NAND

The logical NAND is a logical 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 NAND q (also written as p | q or p ↑ q) is as follows:

Logical NAND
p q p ↑ q
F F T
F T T
T F T
T T F

## Logical NNOR

The logical NNOR is a logical 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 NNOR q (also written as p ⊥ q or p ↓ q) is as follows:

Logical NNOR
p q p ↓ q
F F T
F T F
T F F
T T F

<p>

## 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.