Extension (predicate logic)

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The extension of a
predicate Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, o ...
a truth-valued functionis the set of
tuple In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
s of values that, used as arguments, satisfy the predicate. Such a set of tuples is a relation.

# Examples

For example, the statement "''d2'' is the weekday following ''d1''" can be seen as a truth function associating to each tuple (''d2'', ''d1'') the value ''true'' or ''false''. The extension of this truth function is, by convention, the set of all such tuples associated with the value ''true'', i.e. By examining this extension we can conclude that "Tuesday is the weekday following Saturday" (for example) is false. Using
set-builder notation In set theory and its applications to logic, mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and the ...
, the extension of the ''n''-ary predicate $\Phi$ can be written as :$\\,.$

# Relationship with characteristic function

If the values 0 and 1 in the range of a
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the Function (mathematics), function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'' ...
are identified with the values false and true, respectivelymaking the characteristic function a predicate, then for all relations ''R'' and predicates $\Phi$ the following two statements are equivalent: *$\Phi$ is the characteristic function of ''R'' *''R'' is the extension of $\Phi$