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A '''truth table''' is a tabular array that illustrates the computation of a [[boolean function]], that is, a function of the form <math>f : \mathbb{B}^k \to \mathbb{B},</math> where <math>k\!</math> is a non-negative integer and <math>\mathbb{B}</math> is the [[boolean domain]] <math>\{ 0, 1 \}.\!</math>
<font size="3">☞</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].
A '''truth table''' is a tabular array that illustrates the computation of a ''logical function'', that is, a function of the form <math>f : \mathbb{A}^k \to \mathbb{A},</math> where <math>k\!</math> is a non-negative integer and <math>\mathbb{A}</math> is the domain of logical values <math>\{ \operatorname{false}, \operatorname{true} \}.</math> The names of the logical values, or ''truth values'', are commonly abbreviated in accord with the equations <math>\operatorname{F} = \operatorname{false}</math> and <math>\operatorname{T} = \operatorname{true}.</math>
In many applications it is usual to represent a truth function by a [[boolean function]], that is, a function of the form <math>f : \mathbb{B}^k \to \mathbb{B},</math> where <math>k\!</math> is a non-negative integer and <math>\mathbb{B}</math> is the [[boolean domain]] <math>\{ 0, 1 \}.\!</math> In most applications <math>\operatorname{false}</math> is represented by <math>0\!</math> and <math>\operatorname{true}</math> is represented by <math>1\!</math> 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 <math>\operatorname{F} = 0</math> and <math>\operatorname{T} = 1</math> for granted.
==Logical negation==
==Logical negation==
''[[Logical negation]]'' is an [[logical operation|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.
'''[[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:
The truth table of <math>\operatorname{NOT}~ p,</math> also written <math>\lnot p,\!</math> appears below:
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:40%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%"
|+ '''Logical Negation'''
|+ style="height:30px" | <math>\text{Logical Negation}\!</math>
|- style="background:#e6e6ff"
|- style="height:40px; background:#f0f0ff"
| style="width:50%" | <math>p\!</math>
| style="width:50%" | <math>\lnot p\!</math>
|-
|-
| F || T
| <math>\operatorname{F}</math> || <math>\operatorname{T}</math>
|-
|-
| T || F
| <math>\operatorname{T}</math> || <math>\operatorname{F}</math>
|}
|}
<br>
<br>
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:
The negation of a proposition <math>p\!</math> may be found notated in various ways in various contexts of application, often merely for typographical convenience. Among these variants are the following:
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; width:40%"
{| align="center" border="1" cellpadding="8" cellspacing="0" width="45%"
|+ '''Variant Notations'''
|+ style="height:30px" | <math>\text{Variant Notations}\!</math>
|- style="background:#e6e6ff"
|- style="height:40px; background:#f0f0ff"
| width="50%" align="center" | <math>\text{Notation}\!</math>
! Vocalization
| width="50%" | <math>\text{Vocalization}\!</math>
|-
|-
| style="text-align:center" | <math>\bar{p}</math>
| align="center" | <math>\bar{p}\!</math>
| bar ''p''
| <math>p\!</math> bar
|-
|-
| style="text-align:center" | <math>p'\!</math>
| align="center" | <math>\tilde{p}\!</math>
| ''p'' prime,<p> ''p'' complement
| <math>p\!</math> tilde
|-
|-
| style="text-align:center" | <math>!p\!</math>
| align="center" | <math>p'\!</math>
| bang ''p''
| <math>p\!</math> prime<br> <math>p\!</math> complement
|-
| align="center" | <math>!p\!</math>
| bang <math>p\!</math>
|}
|}
==Logical conjunction==
==Logical conjunction==
''[[Logical conjunction]]'' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both of its operands are true.
'''[[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<math>\cdot</math>q''') is as follows:
The truth table of <math>p ~\operatorname{AND}~ q,</math> also written <math>p \land q\!</math> or <math>p \cdot q,\!</math> appears below:
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%"
|+ '''Logical Conjunction'''
|+ style="height:30px" | <math>\text{Logical Conjunction}\!</math>
|- style="background:#e6e6ff"
|- style="height:40px; background:#f0f0ff"
| style="width:33%" | <math>p\!</math>
| style="width:33%" | <math>q\!</math>
| style="width:33%" | <math>p \land q</math>
|-
|-
| F || F || F
| <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math>
|-
|-
| F || T || F
| <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math>
|-
|-
| T || F || F
| <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math>
|-
|-
| T || T || T
| <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math>
|}
|}
==Logical disjunction==
==Logical disjunction==
''[[Logical disjunction]]'', also called ''logical alternation'', is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if both of its operands are false.
'''[[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:
The truth table of <math>p ~\operatorname{OR}~ q,</math> also written <math>p \lor q,\!</math> appears below:
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%"
|+ '''Logical Disjunction'''
|+ style="height:30px" | <math>\text{Logical Disjunction}\!</math>
|- style="background:#e6e6ff"
|- style="height:40px; background:#f0f0ff"
| style="width:33%" | <math>p\!</math>
| style="width:33%" | <math>q\!</math>
| style="width:33%" | <math>p \lor q</math>
|-
|-
| F || F || F
| <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math>
|-
|-
| F || T || T
| <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math>
|-
|-
| T || F || T
| <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math>
|-
|-
| T || T || T
| <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math>
|}
|}
==Logical equality==
==Logical equality==
''[[Logical equality]]'' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both operands are false or both operands are true.
'''[[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:
The truth table of <math>p ~\operatorname{EQ}~ q,</math> also written <math>p = q,\!</math> <math>p \Leftrightarrow q,\!</math> or <math>p \equiv q,\!</math> appears below:
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%"
|+ '''Logical Equality'''
|+ style="height:30px" | <math>\text{Logical Equality}\!</math>
|- style="background:#e6e6ff"
|- style="height:40px; background:#f0f0ff"
| style="width:33%" | <math>p\!</math>
| style="width:33%" | <math>q\!</math>
| style="width:33%" | <math>p = q\!</math>
|-
|-
| F || F || T
| <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math>
|-
|-
| F || T || F
| <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math>
|-
|-
| T || F || F
| <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math>
|-
|-
| T || T || T
| <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math>
|}
|}
==Exclusive disjunction==
==Exclusive disjunction==
''[[Exclusive disjunction]]'', also known as ''logical inequality'' or ''symmetric difference'', is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' just in case exactly one of its operands is true.
'''[[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:
The truth table of <math>p ~\operatorname{XOR}~ q,</math> also written <math>p + q\!</math> or <math>p \ne q,\!</math> appears below:
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%"
|+ '''Exclusive Disjunction'''
|+ style="height:30px" | <math>\text{Exclusive Disjunction}\!</math>
|- style="background:#e6e6ff"
|- style="height:40px; background:#f0f0ff"
| style="width:33%" | <math>p\!</math>
| style="width:33%" | <math>q\!</math>
| style="width:33%" | <math>p ~\operatorname{XOR}~ q</math>
|-
|-
| F || F || F
| <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math>
|-
|-
| F || T || T
| <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math>
|-
|-
| T || F || T
| <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math>
|-
|-
| T || T || F
| <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math>
|}
|}
<br>
<br>
The following equivalents can then be deduced:
The following equivalents may then be deduced:
{| align="center" cellspacing="10" width="90%"
p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\
|
\\
<math>\begin{matrix}
& = & (p \lor q) & \land & (\lnot p \lor \lnot q) \\
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)
& = & (p \lor q) & \land & \lnot (p \land q)
\end{matrix}</math>
\end{matrix}</math>
|}
==Logical implication==
==Logical implication==
The ''[[logical implication]]'' and the ''[[material conditional]]'' are both associated with an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if the first operand is true and the second operand is false.
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 '''if p then q''' (symbolized as '''p → q''') and the logical implication '''p implies q''' (symbolized as '''p ⇒ q''') is as follows:
The truth table associated with the material conditional <math>\text{if}~ p ~\text{then}~ q,\!</math> symbolized <math>p \rightarrow q,\!</math> and the logical implication <math>p ~\text{implies}~ q,\!</math> symbolized <math>p \Rightarrow q,\!</math> appears below:
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%"
|+ '''Logical Implication'''
|+ style="height:30px" | <math>\text{Logical Implication}\!</math>
|- style="background:#e6e6ff"
|- style="height:40px; background:#f0f0ff"
| style="width:33%" | <math>p\!</math>
| style="width:33%" | <math>q\!</math>
| style="width:33%" | <math>p \Rightarrow q\!</math>
|-
|-
| F || F || T
| <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math>
|-
|-
| F || T || T
| <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math>
|-
|-
| T || F || F
| <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math>
|-
|-
| T || T || T
| <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math>
|}
|}
==Logical NAND==
==Logical NAND==
The ''[[logical NAND]]'' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, 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 '''[[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 NAND q''' (also written as '''p | q''' or '''p ↑ q''') is as follows:
The truth table of <math>p ~\operatorname{NAND}~ q,</math> also written <math>p \stackrel{\circ}{\curlywedge} q\!</math> or <math>p \barwedge q,\!</math> appears below:
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%"
|+ '''Logical NAND'''
|+ style="height:30px" | <math>\text{Logical NAND}\!</math>
|- style="background:#e6e6ff"
|- style="height:40px; background:#f0f0ff"
| style="width:33%" | <math>p\!</math>
| style="width:33%" | <math>q\!</math>
| style="width:33%" | <math>p \stackrel{\circ}{\curlywedge} q\!</math>
|-
|-
| F || F || T
| <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math>
|-
|-
| F || T || T
| <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math>
|-
|-
| T || F || T
| <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math>
|-
|-
| T || T || F
| <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math>
|}
|}
==Logical NNOR==
==Logical NNOR==
The ''[[logical NNOR]]'' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, 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 '''[[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 NNOR q''' (also written as '''p ⊥ q''' or '''p ↓ q''') is as follows:
The truth table of <math>p ~\operatorname{NNOR}~ q,</math> also written <math>p \curlywedge q,\!</math> appears below:
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%"
|+ '''Logical NNOR'''
|+ style="height:30px" | <math>\text{Logical NNOR}\!</math>
|- style="background:#e6e6ff"
|- style="height:40px; background:#f0f0ff"
| style="width:33%" | <math>p\!</math>
| style="width:33%" | <math>q\!</math>
| style="width:33%" | <math>p \curlywedge q\!</math>
|-
|-
| F || F || T
| <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math>
|-
|-
| F || T || F
| <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math>
|-
|-
| T || F || F
| <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math>
|-
|-
| T || T || F
| <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math>
|}
|}
===Focal nodes===
===Focal nodes===
* [[Inquiry Live]]
* [[Inquiry Live]]
* [[Logic Live]]
* [[Logic Live]]
===Peer nodes===
===Peer nodes===
* [http://intersci.ss.uci.edu/wiki/index.php/Truth_table Truth Table @ InterSciWiki]
* [http://mywikibiz.com/Truth_table Truth Table @ MyWikiBiz]
* [http://mywikibiz.com/Truth_table Truth Table @ MyWikiBiz]
* [http://mathweb.org/wiki/Truth_table Truth Table @ MathWeb Wiki]
* [http://ref.subwiki.org/wiki/Truth_table Truth Table @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Truth_table Truth Table @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Truth_table Truth Table @ Wikiversity Beta]
* [http://wiki.oercommons.org/mediawiki/index.php/Truth_table Truth Table @ OER Commons]
* [http://p2pfoundation.net/Truth_Table Truth Table @ P2P Foundation]
===Logical operators===
===Logical operators===
{{col-break}}
{{col-break}}
* [[Inquiry]]
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
* [[Logic of information]]
{{col-break}}
{{col-break}}
{{col-break}}
{{col-break}}
* [[Pragmatic maxim]]
* [[Pragmatic maxim]]
* [[Pragmatic theory of truth]]
* [[Truth theory]]
{{col-end}}
{{col-end}}
===Related articles===
===Related articles===
{{col-begin}}
{{col-break}}
* [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, “Prospects For Inquiry Driven Systems”]
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, “Inquiry Driven Systems : Inquiry Into Inquiry”]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, “Propositional Equation Reasoning Systems”]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, “Differential Logic : Introduction”]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, “Differential Propositional Calculus”]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, “Differential Logic and Dynamic Systems”]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}
==Document history==
==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.
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.
* [http://intersci.ss.uci.edu/wiki/index.php/Truth_table Truth Table], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Truth_table Truth Table], [http://mywikibiz.com/ MyWikiBiz]
* [http://mywikibiz.com/Truth_table Truth Table], [http://mywikibiz.com/ MyWikiBiz]
* [http://semanticweb.org/wiki/Truth_table Truth Table], [http://semanticweb.org/ SemanticWeb]
* [http://semanticweb.org/wiki/Truth_table Truth Table], [http://semanticweb.org/ SemanticWeb]
* [http://wikinfo.org/w/index.php/Truth_table Truth Table], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Truth_table Truth Table], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Truth_table Truth Table], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://beta.wikiversity.org/wiki/Truth_table Truth Table], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Truth_table&oldid=77110085 Truth Table], [http://en.wikipedia.org/ Wikipedia]
* [http://en.wikipedia.org/w/index.php?title=Truth_table&oldid=77110085 Truth Table], [http://en.wikipedia.org/ Wikipedia]
[[Category:Inquiry]]
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