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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 <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> | | 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> |
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− | 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 the lo<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. | + | 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. |
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| ==Logical negation== | | ==Logical negation== |
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| ==Logical NAND== | | ==Logical NAND== |
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− | 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. |
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− | 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: |
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| <br> | | <br> |
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− | {| 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:15%" | p
| + | | style="width:33%" | <math>p\!</math> |
− | ! style="width:15%" | q
| + | | style="width:33%" | <math>q\!</math> |
− | ! style="width:15%" | p ↑ q
| + | | 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> |
| |} | | |} |
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| ==Logical NNOR== | | ==Logical NNOR== |
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− | 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. |
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− | 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: |
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| <br> | | <br> |
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− | {| 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:15%" | p
| + | | style="width:33%" | <math>p\!</math> |
− | ! style="width:15%" | q
| + | | style="width:33%" | <math>q\!</math> |
− | ! style="width:15%" | p ↓ q
| + | | 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> |
| |} | | |} |
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| ===Focal nodes=== | | ===Focal nodes=== |
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− | {{col-begin}}
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− | {{col-break}}
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| * [[Inquiry Live]] | | * [[Inquiry Live]] |
− | {{col-break}}
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| * [[Logic Live]] | | * [[Logic Live]] |
− | {{col-end}}
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| ===Peer nodes=== | | ===Peer nodes=== |
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− | {{col-begin}}
| + | * [http://intersci.ss.uci.edu/wiki/index.php/Truth_table Truth Table @ InterSciWiki] |
− | {{col-break}}
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| * [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]
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− | * [http://netknowledge.org/wiki/Truth_table Truth Table @ NetKnowledge]
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− | * [http://wiki.oercommons.org/mediawiki/index.php/Truth_table Truth Table @ OER Commons]
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− | {{col-break}}
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− | * [http://p2pfoundation.net/Truth_Table Truth Table @ P2P Foundation]
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− | * [http://semanticweb.org/wiki/Truth_table Truth Table @ SemanticWeb]
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| * [http://ref.subwiki.org/wiki/Truth_table Truth Table @ Subject Wikis] | | * [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://beta.wikiversity.org/wiki/Truth_table Truth Table @ Wikiversity Beta] |
− | {{col-end}}
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| ===Logical operators=== | | ===Logical operators=== |
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| ===Related articles=== | | ===Related articles=== |
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− | * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, “Semiotic Information”] | + | {{col-begin}} |
− | | + | {{col-break}} |
− | * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, “Introduction To 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/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, “Prospects For Inquiry Driven Systems”] | + | * [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/Inquiry_Driven_Systems Jon Awbrey, “Inquiry Driven Systems : Inquiry Into Inquiry”] | + | * [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/Propositional_Equation_Reasoning_Systems Jon Awbrey, “Propositional Equation Reasoning Systems”] | + | * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] |
− | | + | {{col-break}} |
− | * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, “Differential Logic : Introduction”] | + | * [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://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, “Differential Propositional Calculus”] | + | * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] |
− | | + | {{col-end}} |
− | * [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, “Differential Logic and Dynamic Systems”] | |
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| ==Document history== | | ==Document history== |
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| 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. |
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− | {{col-begin}}
| + | * [http://intersci.ss.uci.edu/wiki/index.php/Truth_table Truth Table], [http://intersci.ss.uci.edu/ InterSciWiki] |
− | {{col-break}}
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| * [http://mywikibiz.com/Truth_table Truth Table], [http://mywikibiz.com/ MyWikiBiz] | | * [http://mywikibiz.com/Truth_table Truth Table], [http://mywikibiz.com/ MyWikiBiz] |
− | * [http://mathweb.org/wiki/Truth_table Truth Table], [http://mathweb.org/ MathWeb Wiki]
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− | * [http://netknowledge.org/wiki/Truth_table Truth Table], [http://netknowledge.org/ NetKnowledge]
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− | * [http://wiki.oercommons.org/mediawiki/index.php/Truth_table Truth Table], [http://wiki.oercommons.org/ OER Commons]
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− | {{col-break}}
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− | * [http://p2pfoundation.net/Truth_Table Truth Table], [http://p2pfoundation.net/ P2P Foundation]
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| * [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://getwiki.net/-Truth_Table Truth Table], [http://getwiki.net/ GetWiki]
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− | {{col-break}}
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− | * [http://wikinfo.org/index.php/Truth_table Truth Table], [http://wikinfo.org/ Wikinfo]
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− | * [http://textop.org/wiki/index.php?title=Truth_table Truth Table], [http://textop.org/wiki/ Textop Wiki]
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| * [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] |
− | {{col-end}}
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− | <br><sharethis />
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| [[Category:Inquiry]] | | [[Category:Inquiry]] |