Jon Awbrey at 18:31, 7 November 2015

2015-11-07T18:31:02Z

Jon Awbrey: + article

2010-06-22T19:14:38Z

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☞ 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 [[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>

==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.

The truth table of '''NOT p''' (also written as '''~p''' or '''¬p''') is as follows:

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:40%"
|+ '''Logical Negation'''
|- style="background:#e6e6ff"
! style="width:20%" | p
! style="width:20%" | ¬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:

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; width:40%"
|+ '''Variant Notations'''
|- style="background:#e6e6ff"
! style="text-align:center" | Notation
! Vocalization
|-
| style="text-align:center" | <math>\bar{p}</math>
| bar ''p''
|-
| style="text-align:center" | <math>p'\!</math>
| ''p'' prime, ''p'' complement
|-
| style="text-align:center" | <math>!p\!</math>
| bang ''p''
|}

 

==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.

The truth table of '''p AND q''' (also written as '''p &and; q''', '''p & q''', or '''p<math>\cdot</math>q''') is as follows:

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
|+ '''Logical Conjunction'''
|- style="background:#e6e6ff"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | p &and; q
|-
| F || F || F
|-
| F || T || F
|-
| T || F || F
|-
| T || T || T
|}

 

==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.

The truth table of '''p OR q''' (also written as '''p &or; q''') is as follows:

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
|+ '''Logical Disjunction'''
|- style="background:#e6e6ff"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | p &or; q
|-
| F || F || F
|-
| F || T || T
|-
| T || F || T
|-
| T || T || T
|}

 

==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.

The truth table of '''p EQ q''' (also written as '''p = q''', '''p ↔ q''', or '''p &equiv; q''') is as follows:

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
|+ '''Logical Equality'''
|- style="background:#e6e6ff"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | 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 [[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.

The truth table of '''p XOR q''' (also written as '''p + q''', '''p &oplus; q''', or '''p ≠ q''') is as follows:

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
|+ '''Exclusive Disjunction'''
|- style="background:#e6e6ff"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | p XOR q
|-
| F || F || F
|-
| F || T || T
|-
| T || F || T
|-
| T || T || F
|}

 

The following equivalents can then be deduced:

: <math>\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}</math>

==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 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:

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
|+ '''Logical Implication'''
|- style="background:#e6e6ff"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | 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 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 truth table of '''p NAND q''' (also written as '''p | q''' or '''p ↑ q''') is as follows:

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
|+ '''Logical NAND'''
|- style="background:#e6e6ff"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | 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 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 truth table of '''p NNOR q''' (also written as '''p &perp; q''' or '''p ↓ q''') is as follows:

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
|+ '''Logical NNOR'''
|- style="background:#e6e6ff"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | p ↓ q
|-
| F || F || T
|-
| F || T || F
|-
| T || F || F
|-
| T || T || F
|}

 

==Translations==

* [http://zh.wikipedia.org/wiki/%E7%9C%9F%E5%80%BC%E8%A1%A8 中文 : 真值表]

==Syllabus==

===Focal nodes===

{{col-begin}}
{{col-break}}
* [[Inquiry Live]]
{{col-break}}
* [[Logic Live]]
{{col-end}}

===Peer nodes===

{{col-begin}}
{{col-break}}
* [http://mywikibiz.com/Truth_table Truth Table @ MyWikiBiz]
* [http://mathweb.org/wiki/Truth_table Truth Table @ MathWeb Wiki]
* [http://netknowledge.org/wiki/Truth_table Truth Table @ NetKnowledge]
* [http://wiki.oercommons.org/mediawiki/index.php/Truth_table Truth Table @ OER Commons]
{{col-break}}
* [http://p2pfoundation.net/Truth_Table Truth Table @ P2P Foundation]
* [http://semanticweb.org/wiki/Truth_table Truth Table @ SemanticWeb]
* [http://ref.subwiki.org/wiki/Truth_table Truth Table @ Subject Wikis]
* [http://beta.wikiversity.org/wiki/Truth_table Truth Table @ Wikiversity Beta]
{{col-end}}

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, “Semiotic Information”]

* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, “Introduction To Inquiry Driven Systems”]

* [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, “Prospects For Inquiry Driven Systems”]

* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, “Inquiry Driven Systems : Inquiry Into Inquiry”]

* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, “Propositional Equation Reasoning Systems”]

* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, “Differential Logic : Introduction”]

* [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, “Differential Propositional Calculus”]

* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, “Differential Logic and Dynamic Systems”]

==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.

{{col-begin}}
{{col-break}}
* [http://mywikibiz.com/Truth_table Truth Table], [http://mywikibiz.com/ MyWikiBiz]
* [http://mathweb.org/wiki/Truth_table Truth Table], [http://mathweb.org/ MathWeb Wiki]
* [http://netknowledge.org/wiki/Truth_table Truth Table], [http://netknowledge.org/ NetKnowledge]
* [http://wiki.oercommons.org/mediawiki/index.php/Truth_table Truth Table], [http://wiki.oercommons.org/ OER Commons]
{{col-break}}
* [http://p2pfoundation.net/Truth_Table Truth Table], [http://p2pfoundation.net/ P2P Foundation]
* [http://semanticweb.org/wiki/Truth_table Truth Table], [http://semanticweb.org/ SemanticWeb]
* [http://beta.wikiversity.org/wiki/Truth_table Truth Table], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://getwiki.net/-Truth_Table Truth Table], [http://getwiki.net/ GetWiki]
{{col-break}}
* [http://wikinfo.org/index.php/Truth_table Truth Table], [http://wikinfo.org/ Wikinfo]
* [http://textop.org/wiki/index.php?title=Truth_table Truth Table], [http://textop.org/wiki/ Textop Wiki]
* [http://en.wikipedia.org/w/index.php?title=Truth_table&oldid=77110085 Truth Table], [http://en.wikipedia.org/ Wikipedia]
{{col-end}}

[[Category:Inquiry]]
[[Category:Open Educational Resource]]
[[Category:Peer Educational Resource]]
[[Category:Combinatorics]]
[[Category:Computer Science]]
[[Category:Discrete Mathematics]]
[[Category:Formal Languages]]
[[Category:Formal Sciences]]
[[Category:Formal Systems]]
[[Category:Linguistics]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Philosophy]]
[[Category:Semiotics]]

Truth table - Revision history

Jon Awbrey at 18:31, 7 November 2015

Jon Awbrey: + article