Definition:Set/Definition by Predicate

Definition
An object can be specified by means of a predicate, that is, in terms of a property (or properties) that it possesses.

Whether an object $x$ possesses a particular property $P$ is either true or false (in Aristotelian logic) and so can be the subject of a propositional function $\map P x$.

Hence a set can be specified by means of such a propositional function:
 * $S = \set {x: \map P x}$

which means:
 * $S$ is the set of all objects which have the property $P$

or, more formally:
 * $S$ is the set of all $x$ such that $\map P x$ is true.

In this context, we see that the symbol $:$ is interpreted as such that.

Also known as
This construction is sometimes known as the set-builder notation or as set comprehension.

This is also sometimes rendered as set builder notation.

An alternative notation for this is $S = \set {x \mid \map P x}$, but it can be argued that the use of $\mid$ for such that can cause ambiguity and confusion, as $\mid$ has several other meanings in mathematics.

On the other hand, if the expression defining the predicate is thick with $:$ characters, it may improve clarity to use $\mid$ for such that after all.

Some authors, mindful of such confusion, use the notation $S = \set {x; \map P x}$ as the semicolon is relatively rare in mathematical notation.

Sometimes it is convenient to abbreviate the notation by simply writing $S = \set {\map P x}$ or even just $S = \set P$.

For example, to describe the set $\set {x \in \R: \map f x \le \map g x}$ (for appropriate functions $f, g$), one could simply use $\set {f \le g}$.

Some sources simply identify $x$ as a variable, and then refer to $A = \set x$ as the set of all the values that $x$ can take.

Some sources use the notation $\boldsymbol [x: \map P x \boldsymbol ]$ for $\set {\map P x}$.

Axiomatic Set Theory
In the context of axiomatic set theory, a more strictly rigorous presentation of this concept is:
 * $S = \set {x \in A: \map P x}$

which means:
 * $S$ is the set of all objects in $A$ which have the property $P$

or, more formally:
 * $S$ is the set of all $x$ in $A$ such that $\map P x$ is true.

This presupposes that all the objects under consideration for inclusion in $S$ already belong to some previously-defined set $A$.

Thus any set $S$ can be expressed as:
 * $S = \set {s: s \in S}$

See the Axiom of Specification.

Also see

 * Definition:Explicit Set Definition
 * Definition:Implicit Set Definition


 * Definition:Class-Builder Notation