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Logical equality

Logical equality is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false or both operands are true.

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

• $p\Leftrightarrow q$ • $p\equiv q$ • $p=q$ A truth table for $p=q$ appears below:

 $p$ $q$ $p=q$ $\mathrm {F}$ $\mathrm {F}$ $\mathrm {T}$ $\mathrm {F}$ $\mathrm {T}$ $\mathrm {F}$ $\mathrm {T}$ $\mathrm {F}$ $\mathrm {F}$ $\mathrm {T}$ $\mathrm {T}$ $\mathrm {T}$ A logical graph for $p=q$ is shown below:

The traversal string of this graph is ${\texttt {((}}p{\texttt {,}}q{\texttt {))}}.$ The proposition $p=q$ 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 $p=q$ indicates the region where $p=q$ is true by means of a distinctive color or shading.  In this case the region consists of two single cells, 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.