Logical NNOR

Logical NNOR (“Neither Nor”) is an operation on two logical values, typically the values of two propositions, 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.

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

• $p\curlywedge q$
• $p~{\bar {\lor }}~q$
• $p\downarrow q$

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

${\text{Logical NNOR}}$

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

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

The traversal string of this graph is ${\texttt {(}}p{\texttt {)(}}q{\texttt {)}}.$  The proposition $p\curlywedge q$ 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 $p\curlywedge q$ indicates the region where $p\curlywedge q$ is true by means of a distinctive color or shading.  In this case the region is a single cell, 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 NNOR, InterSciWiki
• Logical NNOR, Wikiversity
• Logical NNOR, Wikipedia