Logical conjunction

Logical conjunction is an operation on two logical values, typically the values of two propositions, which produces a value of true if and only if both of its operands are true.

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

• $p~\mathrm {and} ~q$
• $p\land q$
• $p\cdot q$
• $p~q$
• $pq$

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

${\text{Logical Conjunction}}$

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

A logical graph for $p\land q$ is drawn as two letters attached to a root node:

Written as a string, this is just the concatenation $pq.$  The proposition $pq$ 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 $p\land q$ indicates the region where $pq$ 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 Conjunction, InterSciWiki
• Logical Conjunction, Wikiversity
• Logical Conjunction, Wikipedia