# 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 $p$ 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:

 $p$ $q$ $p\land q$ $\mathrm {F}$ $\mathrm {F}$ $\mathrm {F}$ $\mathrm {F}$ $\mathrm {T}$ $\mathrm {F}$ $\mathrm {T}$ $\mathrm {F}$ $\mathrm {F}$ $\mathrm {T}$ $\mathrm {T}$ $\mathrm {T}$ 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: ## Document history

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