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# 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 ${\displaystyle p}$ and ${\displaystyle q}$ may be written in various ways.  Among the most common are these:

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

A truth table for ${\displaystyle p+q}$ appears below:

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

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

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

A logical graph for ${\displaystyle p+q}$ is shown below:

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

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

## 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.