Difference between revisions of "Truth table"

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<font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].
 
<font size="3">&#9758;</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 '''truth 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 '''truth value domain''' <math>\{ \operatorname{false}, \operatorname{true} \},</math> commonly abbreviated as <math>\{ \operatorname{F}, \operatorname{T} \}.</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>
  
 
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.
 
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==
  
−
''[[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 '''&not;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:20%" | p
+
| style="width:50%" | <math>p\!</math>
−
! style="width:20%" | &not;p
+
| 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"
−
! style="text-align:center" | Notation
+
| 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>
 
|}
 
|}
  
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==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 &and; 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:15%" | p
+
| style="width:33%" | <math>p\!</math>
−
! style="width:15%" | q
+
| style="width:33%" | <math>q\!</math>
−
! style="width:15%" | p &and; q
+
| 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>
 
|}
 
|}
  
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==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 &or; 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:15%" | p
+
| style="width:33%" | <math>p\!</math>
−
! style="width:15%" | q
+
| style="width:33%" | <math>q\!</math>
−
! style="width:15%" | p &or; q
+
| 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>
 
|}
 
|}
  
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==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 &harr; q''', or '''p &equiv; 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: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 = 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>
 
|}
 
|}
  
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==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 &oplus; q''', or '''p &ne; 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:15%" | p
+
| style="width:33%" | <math>p\!</math>
−
! style="width:15%" | q
+
| style="width:33%" | <math>q\!</math>
−
! style="width:15%" | p XOR q
+
| 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:
  
−
: <math>\begin{matrix}
+
{| 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&nbsp;&rarr;&nbsp;q''') and the logical implication '''p implies q''' (symbolized as '''p&nbsp;&rArr;&nbsp;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:15%" | p
+
| style="width:33%" | <math>p\!</math>
−
! style="width:15%" | q
+
| style="width:33%" | <math>q\!</math>
−
! style="width:15%" | p &rArr; q
+
| 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>
 
|}
 
|}
  
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==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&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>
 
<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>
 
|}
 
|}
  
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==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]]''' (&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>
 
<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>
 
|}
 
|}
  
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===Focal nodes===
 
===Focal nodes===
  
−
{{col-begin}}
 
−
{{col-break}}
 
 
* [[Inquiry Live]]
 
* [[Inquiry Live]]
−
{{col-break}}
 
 
* [[Logic Live]]
 
* [[Logic Live]]
−
{{col-end}}
 
  
 
===Peer nodes===
 
===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://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://en.wikiversity.org/wiki/Truth_table Truth Table @ Wikiversity]
 
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* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
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* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
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* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
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* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;]
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* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
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* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;]
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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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* [http://p2pfoundation.net/Truth_Table Truth Table], [http://p2pfoundation.net/ P2P Foundation]
 
 
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* [http://wikinfo.org/w/index.php/Truth_table Truth Table], [http://wikinfo.org/w/ Wikinfo]
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* [http://en.wikiversity.org/wiki/Truth_table Truth Table], [http://en.wikiversity.org/ Wikiversity]
 
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[[Category:Inquiry]]
 
[[Category:Inquiry]]

Latest revision as of 03:25, 30 October 2015

☞ This page belongs to resource collections on Logic and Inquiry.

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

Syllabus

Focal nodes

Peer nodes

Logical operators

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Relational concepts

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Information, Inquiry

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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.

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