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Truth table (view source)

Revision as of 15:10, 26 May 2009

204 bytes added ,  15:10, 26 May 2009
table colors → table body (#f8f8ff = ghostwhite) table head (#e6e6ff = blue gray)
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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 ''f'' : '''B'''<sup>''k''</sup> &rarr; '''B''', where ''k'' is a non-negative integer and '''B''' is the [[boolean domain]] {0,&nbsp;1}.
+
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==
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The truth table of '''NOT p''' (also written as '''~p''' or '''&not;p''') is as follows:
 
The truth table of '''NOT p''' (also written as '''~p''' or '''&not;p''') is as follows:
   −
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:40%"
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<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:40%"
 
|+ '''Logical Negation'''
 
|+ '''Logical Negation'''
−
|- style="background:paleturquoise"
+
|- style="background:#e6e6ff"
 
! style="width:20%" | p
 
! style="width:20%" | p
 
! style="width:20%" | &not;p
 
! style="width:20%" | &not;p
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| T || F
 
| T || F
 
|}
 
|}
 +
 
<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 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:
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; width:40%"
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<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; width:40%"
 
|+ '''Variant Notations'''
 
|+ '''Variant Notations'''
−
|- style="background:paleturquoise"
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|- style="background:#e6e6ff"
 
! style="text-align:center" | Notation
 
! style="text-align:center" | Notation
 
! Vocalization
 
! Vocalization
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| bang ''p''
 
| bang ''p''
 
|}
 
|}
 +
 
<br>
 
<br>
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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 '''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:lightcyan; font-weight:bold; text-align:center; width:45%"
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<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
 
|+ '''Logical Conjunction'''
 
|+ '''Logical Conjunction'''
−
|- style="background:paleturquoise"
+
|- style="background:#e6e6ff"
 
! style="width:15%" | p
 
! style="width:15%" | p
 
! style="width:15%" | q
 
! style="width:15%" | q
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| T || T || T
 
| T || T || T
 
|}
 
|}
 +
 
<br>
 
<br>
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The truth table of '''p OR q''' (also written as '''p &or; q''') is as follows:
 
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:lightcyan; font-weight:bold; text-align:center; width:45%"
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<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
 
|+ '''Logical Disjunction'''
 
|+ '''Logical Disjunction'''
−
|- style="background:paleturquoise"
+
|- style="background:#e6e6ff"
 
! style="width:15%" | p
 
! style="width:15%" | p
 
! style="width:15%" | q
 
! style="width:15%" | q
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| T || T || T
 
| T || T || T
 
|}
 
|}
 +
 
<br>
 
<br>
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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 '''p EQ q''' (also written as '''p = q''', '''p &harr; q''', or '''p &equiv; q''') is as follows:
   −
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
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<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
 
|+ '''Logical Equality'''
 
|+ '''Logical Equality'''
−
|- style="background:paleturquoise"
+
|- style="background:#e6e6ff"
 
! style="width:15%" | p
 
! style="width:15%" | p
 
! style="width:15%" | q
 
! style="width:15%" | q
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| T || T || T
 
| T || T || T
 
|}
 
|}
 +
 
<br>
 
<br>
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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 '''p XOR q''' (also written as '''p + q''', '''p &oplus; q''', or '''p &ne; q''') is as follows:
   −
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
+
<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
 
|+ '''Exclusive Disjunction'''
 
|+ '''Exclusive Disjunction'''
−
|- style="background:paleturquoise"
+
|- style="background:#e6e6ff"
 
! style="width:15%" | p
 
! style="width:15%" | p
 
! style="width:15%" | q
 
! style="width:15%" | q
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| T || T || F
 
| T || T || F
 
|}
 
|}
 +
 
<br>
 
<br>
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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 '''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:
   −
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
+
<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
 
|+ '''Logical Implication'''
 
|+ '''Logical Implication'''
−
|- style="background:paleturquoise"
+
|- style="background:#e6e6ff"
 
! style="width:15%" | p
 
! style="width:15%" | p
 
! style="width:15%" | q
 
! style="width:15%" | q
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| T || T || T
 
| T || T || T
 
|}
 
|}
 +
 
<br>
 
<br>
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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 '''p NAND q''' (also written as '''p&nbsp;|&nbsp;q''' or '''p&nbsp;&uarr;&nbsp;q''') is as follows:
   −
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
+
<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
 
|+ '''Logical NAND'''
 
|+ '''Logical NAND'''
−
|- style="background:paleturquoise"
+
|- style="background:#e6e6ff"
 
! style="width:15%" | p
 
! style="width:15%" | p
 
! style="width:15%" | q
 
! style="width:15%" | q
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| T || T || F
 
| T || T || F
 
|}
 
|}
 +
 
<br>
 
<br>
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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 '''p NNOR q''' (also written as '''p&nbsp;&perp;&nbsp;q''' or '''p&nbsp;&darr;&nbsp;q''') is as follows:
   −
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
+
<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
 
|+ '''Logical NNOR'''
 
|+ '''Logical NNOR'''
−
|- style="background:paleturquoise"
+
|- style="background:#e6e6ff"
 
! style="width:15%" | p
 
! style="width:15%" | p
 
! style="width:15%" | q
 
! style="width:15%" | q
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| T || T || F
 
| T || T || F
 
|}
 
|}
 +
 
<br>
 
<br>
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[[Category:Computer Science]]
 
[[Category:Computer Science]]
 
[[Category:Discrete Mathematics]]
 
[[Category:Discrete Mathematics]]
 +
[[Category:Formal Languages]]
 +
[[Category:Formal Sciences]]
 +
[[Category:Formal Systems]]
 +
[[Category:Linguistics]]
 
[[Category:Logic]]
 
[[Category:Logic]]
 
[[Category:Mathematics]]
 
[[Category:Mathematics]]
 +
[[Category:Philosophy]]
 +
[[Category:Semiotics]]
 +
 +
<sharethis />
Jon Awbrey
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