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MyWikiBiz, Author Your Legacy — Saturday May 04, 2024
Jump to navigationJump to searchcopy text from [http://www.opencycle.net/ OpenCycle] of which Jon Awbrey is the sole author
'''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>
==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]]
|}
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>
==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]]
|}
