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MyWikiBiz, Author Your Legacy — Friday September 05, 2025
Jump to navigationJump to searchcopy text from [http://www.opencycle.net/ OpenCycle] of which Jon Awbrey is the sole author
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> → '''B''', where ''k'' is a non-negative integer and '''B''' is the [[boolean domain]] {0, 1}.
==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:lightcyan; font-weight:bold; text-align:center; width:40%"
|+ '''Logical Negation'''
|- style="background:paleturquoise"
! style="width:20%" | p
! style="width:20%" | ¬p
|-
| F || T
|-
| T || F
|}
<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:
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; width:40%"
|+ '''Variant Notations'''
|- style="background:paleturquoise"
! 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> ''p'' complement
|-
| style="text-align:center" | <math>!p\!</math>
| bang ''p''
|}
<br>
==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 ∧ 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%"
|+ '''Logical Conjunction'''
|- style="background:paleturquoise"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | p ∧ q
|-
| F || F || F
|-
| F || T || F
|-
| T || F || F
|-
| T || T || T
|}
<br>
==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 ∨ q''') is as follows:
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
|+ '''Logical Disjunction'''
|- style="background:paleturquoise"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | p ∨ q
|-
| F || F || F
|-
| F || T || T
|-
| T || F || T
|-
| T || T || T
|}
<br>
==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 ≡ q''') is as follows:
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
|+ '''Logical Equality'''
|- style="background:paleturquoise"
! 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
|}
<br>
==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 ⊕ q''', or '''p ≠ q''') is as follows:
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
|+ '''Exclusive Disjunction'''
|- style="background:paleturquoise"
! 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
|}
<br>
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:lightcyan; font-weight:bold; text-align:center; width:45%"
|+ '''Logical Implication'''
|- style="background:paleturquoise"
! 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
|}
<br>
==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:lightcyan; font-weight:bold; text-align:center; width:45%"
|+ '''Logical NAND'''
|- style="background:paleturquoise"
! 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
|}
<br>
==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 ⊥ q''' or '''p ↓ q''') is as follows:
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
|+ '''Logical NNOR'''
|- style="background:paleturquoise"
! 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
|}
<br>
==See also==
===Logical operators===
{|
| valign=top |
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
| valign=top |
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
|}
===Related topics===
{|
| valign=top |
* [[Ampheck]]
* [[Boolean algebra]]
* [[Boolean domain]]
* [[Boolean function]]
| valign=top |
* [[Boolean logic]]
* [[Laws of Form]]
* [[Logic gate]]
* [[Logical graph]]
| valign=top |
* [[Peirce's law]]
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Zeroth order logic]]
|}
==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:lightcyan; font-weight:bold; text-align:center; width:40%"
|+ '''Logical Negation'''
|- style="background:paleturquoise"
! style="width:20%" | p
! style="width:20%" | ¬p
|-
| F || T
|-
| T || F
|}
<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:
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; width:40%"
|+ '''Variant Notations'''
|- style="background:paleturquoise"
! 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> ''p'' complement
|-
| style="text-align:center" | <math>!p\!</math>
| bang ''p''
|}
<br>
==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 ∧ 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%"
|+ '''Logical Conjunction'''
|- style="background:paleturquoise"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | p ∧ q
|-
| F || F || F
|-
| F || T || F
|-
| T || F || F
|-
| T || T || T
|}
<br>
==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 ∨ q''') is as follows:
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
|+ '''Logical Disjunction'''
|- style="background:paleturquoise"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | p ∨ q
|-
| F || F || F
|-
| F || T || T
|-
| T || F || T
|-
| T || T || T
|}
<br>
==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 ≡ q''') is as follows:
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
|+ '''Logical Equality'''
|- style="background:paleturquoise"
! 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
|}
<br>
==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 ⊕ q''', or '''p ≠ q''') is as follows:
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
|+ '''Exclusive Disjunction'''
|- style="background:paleturquoise"
! 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
|}
<br>
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:lightcyan; font-weight:bold; text-align:center; width:45%"
|+ '''Logical Implication'''
|- style="background:paleturquoise"
! 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
|}
<br>
==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:lightcyan; font-weight:bold; text-align:center; width:45%"
|+ '''Logical NAND'''
|- style="background:paleturquoise"
! 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
|}
<br>
==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 ⊥ q''' or '''p ↓ q''') is as follows:
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
|+ '''Logical NNOR'''
|- style="background:paleturquoise"
! 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
|}
<br>
==See also==
===Logical operators===
{|
| valign=top |
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
| valign=top |
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
|}
===Related topics===
{|
| valign=top |
* [[Ampheck]]
* [[Boolean algebra]]
* [[Boolean domain]]
* [[Boolean function]]
| valign=top |
* [[Boolean logic]]
* [[Laws of Form]]
* [[Logic gate]]
* [[Logical graph]]
| valign=top |
* [[Peirce's law]]
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Zeroth order logic]]
|}
