Difference between revisions of "Truth table"

Latest revision as of 03:25, 30 October 2015

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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 \(f : \mathbb{A}^k \to \mathbb{A},\) 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:

\(\text{Logical Negation}\!\)
\(p\!\)	\(\lnot p\!\)
\(\operatorname{F}\)	\(\operatorname{T}\)
\(\operatorname{T}\)	\(\operatorname{F}\)

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:

\(\text{Variant Notations}\!\)
\(\text{Notation}\!\)	\(\text{Vocalization}\!\)
\(\bar{p}\!\)	\(p\!\) 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:

\(\text{Logical Conjunction}\!\)
\(p\!\)	\(q\!\)	\(p \land q\)
\(\operatorname{F}\)	\(\operatorname{F}\)	\(\operatorname{F}\)
\(\operatorname{F}\)	\(\operatorname{T}\)	\(\operatorname{F}\)
\(\operatorname{T}\)	\(\operatorname{F}\)	\(\operatorname{F}\)
\(\operatorname{T}\)	\(\operatorname{T}\)	\(\operatorname{T}\)

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:

\(\text{Logical Disjunction}\!\)
\(p\!\)	\(q\!\)	\(p \lor q\)
\(\operatorname{F}\)	\(\operatorname{F}\)	\(\operatorname{F}\)
\(\operatorname{F}\)	\(\operatorname{T}\)	\(\operatorname{T}\)
\(\operatorname{T}\)	\(\operatorname{F}\)	\(\operatorname{T}\)
\(\operatorname{T}\)	\(\operatorname{T}\)	\(\operatorname{T}\)

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:

\(\text{Logical Equality}\!\)
\(p\!\)	\(q\!\)	\(p = q\!\)
\(\operatorname{F}\)	\(\operatorname{F}\)	\(\operatorname{T}\)
\(\operatorname{F}\)	\(\operatorname{T}\)	\(\operatorname{F}\)
\(\operatorname{T}\)	\(\operatorname{F}\)	\(\operatorname{F}\)
\(\operatorname{T}\)	\(\operatorname{T}\)	\(\operatorname{T}\)

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:

\(\text{Exclusive Disjunction}\!\)
\(p\!\)	\(q\!\)	\(p ~\operatorname{XOR}~ q\)
\(\operatorname{F}\)	\(\operatorname{F}\)	\(\operatorname{F}\)
\(\operatorname{F}\)	\(\operatorname{T}\)	\(\operatorname{T}\)
\(\operatorname{T}\)	\(\operatorname{F}\)	\(\operatorname{T}\)
\(\operatorname{T}\)	\(\operatorname{T}\)	\(\operatorname{F}\)

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:

\(\text{Logical Implication}\!\)
\(p\!\)	\(q\!\)	\(p \Rightarrow q\!\)
\(\operatorname{F}\)	\(\operatorname{F}\)	\(\operatorname{T}\)
\(\operatorname{F}\)	\(\operatorname{T}\)	\(\operatorname{T}\)
\(\operatorname{T}\)	\(\operatorname{F}\)	\(\operatorname{F}\)
\(\operatorname{T}\)	\(\operatorname{T}\)	\(\operatorname{T}\)

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:

\(\text{Logical NAND}\!\)
\(p\!\)	\(q\!\)	\(p \stackrel{\circ}{\curlywedge} q\!\)
\(\operatorname{F}\)	\(\operatorname{F}\)	\(\operatorname{T}\)
\(\operatorname{F}\)	\(\operatorname{T}\)	\(\operatorname{T}\)
\(\operatorname{T}\)	\(\operatorname{F}\)	\(\operatorname{T}\)
\(\operatorname{T}\)	\(\operatorname{T}\)	\(\operatorname{F}\)

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:

\(\text{Logical NNOR}\!\)
\(p\!\)	\(q\!\)	\(p \curlywedge q\!\)
\(\operatorname{F}\)	\(\operatorname{F}\)	\(\operatorname{T}\)
\(\operatorname{F}\)	\(\operatorname{T}\)	\(\operatorname{F}\)
\(\operatorname{T}\)	\(\operatorname{F}\)	\(\operatorname{F}\)
\(\operatorname{T}\)	\(\operatorname{T}\)	\(\operatorname{F}\)

Translations

中文 : 真值表

Syllabus

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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.

@@ Line 3: / Line 3: @@
 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 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.
 ==Logical negation==
@@ Line 196: / Line 196: @@
 ==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&nbsp;|&nbsp;q''' or '''p&nbsp;&uarr;&nbsp;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>
-{| 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 &uarr; 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>
 |}
@@ Line 222: / Line 222: @@
 ==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]]''' (&ldquo;Neither Nor&rdquo;) 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&nbsp;&perp;&nbsp;q''' or '''p&nbsp;&darr;&nbsp;q''') is as follows:
+The truth table of <math>p ~\operatorname{NNOR}~ q,</math> also written <math>p \curlywedge q,\!</math> appears below:
 <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:15%" | p
+| style="width:33%" | <math>p\!</math>
-! style="width:15%" | q
+| style="width:33%" | <math>q\!</math>
-! style="width:15%" | p &darr; 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>
 |}
@@ Line 254: / Line 254: @@
 ===Focal nodes===
-{{col-begin}}
-{{col-break}}
 * [[Inquiry Live]]
-{{col-break}}
 * [[Logic Live]]
-{{col-end}}
 ===Peer nodes===
-{{col-begin}}
+* [http://intersci.ss.uci.edu/wiki/index.php/Truth_table Truth Table @ InterSciWiki]
-{{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://en.wikiversity.org/wiki/Truth_table Truth Table @ Wikiversity]
 * [http://beta.wikiversity.org/wiki/Truth_table Truth Table @ Wikiversity Beta]
-{{col-end}}
 ===Logical operators===
@@ Line 353: / Line 342: @@
 ===Related articles===
-* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;]
+{{col-begin}}
+{{col-break}}
-* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;]
+* [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, &ldquo;Prospects For Inquiry Driven Systems&rdquo;]
+* [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, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;]
+* [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, &ldquo;Propositional Equation Reasoning Systems&rdquo;]
+* [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, &ldquo;Differential Logic : Introduction&rdquo;]
+* [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, &ldquo;Differential Propositional Calculus&rdquo;]
+* [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, &ldquo;Differential Logic and Dynamic Systems&rdquo;]
 ==Document history==
@@ Line 373: / Line 361: @@
 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}}
+* [http://intersci.ss.uci.edu/wiki/index.php/Truth_table Truth Table], [http://intersci.ss.uci.edu/ InterSciWiki]
-{{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://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://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}}
-<br><sharethis />
 [[Category:Inquiry]]

Difference between revisions of "Truth table"

Latest revision as of 03:25, 30 October 2015

Contents

Logical negation

Logical conjunction

Logical disjunction

Logical equality

Exclusive disjunction

Logical implication

Logical NAND

Logical NNOR

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Syllabus

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Logical operators

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