Editing Domain of discourse (section)

==Examples==
For example, in an [[interpretation (logic)|interpretation]] of [[first-order logic]], the domain of discourse is the set of individuals over which the [[Quantifier (logic)|quantifiers]] range. A proposition such as {{math|[[Universal quantification|∀]]''x'' (''x''<sup>2</sup> ≠ 2)}} is ambiguous if no domain of discourse has been identified. In one interpretation, the domain of discourse could be the set of [[real number]]s; in another interpretation, it could be the set of [[natural number]]s. If the domain of discourse is the set of real numbers, the proposition is false, with {{math|1=''x'' = {{radic|2}}}} as counterexample; if the domain is the set of natural numbers, the proposition is true, since 2 is not the square of any natural number.

The [[binary relation]] called [[set membership]], expressed as <math> x \in A</math>, and meaning that ''x'' belongs to set ''A'', is clear enough. Every binary relation has a [[converse relation]], and the converse of <math>\in \ \ \text{is written}\ \  \ni</math>. Also, a binary relation must have a '''domain'''. The domain of the converse of set membership is the universe of discourse. Any subset of this universe may, or may not, contain ''x''. ''A'' is a subset of this universe, not necessarily restricted to ''A''.