# Logical conjunction

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

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

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

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

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

Written as a string, this is just the concatenation ${\displaystyle pq.}$  The proposition ${\displaystyle pq}$ 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 ${\displaystyle p\land q}$ indicates the region, in this case a single cell, where ${\displaystyle pq}$ is true by means of a distinct color or shading, 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.
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