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.

An exclusive disjunction of propositions $p$ and $q$ may be written in various ways.  Among the most common are these:

• $p~\mathrm {xor} ~q$
• $p~\Delta ~q$
• $p\neq q$
• $p+q$

A truth table for $p+q$ appears below:

${\text{Exclusive Disjunction}}$

$p$	$q$	$p+q$
$\mathrm {F}$	$\mathrm {F}$	$\mathrm {F}$
$\mathrm {F}$	$\mathrm {T}$	$\mathrm {T}$
$\mathrm {T}$	$\mathrm {F}$	$\mathrm {T}$
$\mathrm {T}$	$\mathrm {T}$	$\mathrm {F}$

The exclusive disjunction of two variables belongs to the family of minimal negation operators.  Thus, we have the following equivalents:

• $p+q$
• $\nu (p,q)$
• ${\texttt {(}}p{\texttt {,}}q{\texttt {)}}$

A logical graph for $p+q$ is shown below:

The traversal string of this graph is ${\texttt {(}}p{\texttt {,}}q{\texttt {)}}.$  The proposition $p+q$ may be taken as a Boolean function $f(p,q)$ having the abstract type $f:\mathbb {B} \times \mathbb {B} \to \mathbb {B} ,$ where $\mathbb {B} =\{0,1\}$ is interpreted in such a way that $0$ means $\mathrm {false}$ and $1$ means $\mathrm {true} .$

A Venn diagram for $p+q$ indicates the region where $p+q$ is true by means of a distinctive color or shading.  In this case the region is two single cells, as shown below:

Resources

• Logic Syllabus

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.

• Exclusive Disjunction, InterSciWiki
• Exclusive Disjunction, Wikiversity
• Exclusive Disjunction, Wikipedia