Logical disjunction

Logical disjunction, also called logical alternation, is an operation on two logical values, typically the values of two propositions, that produces a value of false if and only if both of its operands are false.

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

• $p\mathrm{o}\mathrm{r}q$
• $p\vee q$

A truth table for $p\vee q$ appears below:

$\text{Logical Disjunction}$

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

A logical graph for $p\vee q$ is shown below:

The traversal string of this graph is $\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mo stretchy="false">(</mo></mrow>}p\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">(</mo></mrow>}q\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow>}.$  The proposition $p\vee q$ 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 $\mathrm{f}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{e}$ and $1$ means $\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{e}.$

A Venn diagram for $p\vee q$ indicates the region where $p\vee q$ is true by means of a distinctive color or shading.  In this case the region consists of three adjacent 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.

• Logical Disjunction, InterSciWiki
• Logical Disjunction, Wikiversity
• Logical Disjunction, Wikipedia