Logical conjunction

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Logical conjunction 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 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, in this case a single cell, where $pq$ is true by means of a distinct color or shading, as shown below:

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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.