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-value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (''true'' or '' false'').
Computing
In some progra ...
d
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
is the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of
tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
s of values that, used as arguments, satisfy the predicate. Such a set of tuples is a
relation
Relation or relations may refer to:
General uses
*International relations, the study of interconnection of politics, economics, and law on a global level
*Interpersonal relationship, association or acquaintance between two or more people
*Public ...
.
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, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy.
Defining ...
, the extension of the
''n''-ary predicate
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
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
are identified with the values false and true, respectivelymaking the characteristic function a predicate, then for all relations ''R'' and predicates
the following two statements are equivalent:
*
is the characteristic function of ''R''
*''R'' is the extension of
See also
*
Extensional logic
Intensional logic is an approach to predicate logic that extends first-order logic, which has quantifiers that range over the individuals of a universe (''extensions''), by additional quantifiers that range over terms that may have such individuals ...
*
Extensional set
In mathematics, a setoid (''X'', ~) is a set (or type) ''X'' equipped with an equivalence relation ~. A setoid may also be called E-set, Bishop set, or extensional set.
Setoids are studied especially in proof theory and in type-theoretic fou ...
*
Extensionality
In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned with whether the internal ...
*
Intension
In any of several fields of study that treat the use of signs — for example, in linguistics, logic, mathematics, semantics, semiotics, and philosophy of language — an intension is any property or quality connoted by a word, phrase, or anoth ...
References
extension (semantics)in
nLab
Predicate logic
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