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 (characteristic) 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.

We can express this symbolically as:
 * $\forall x: \paren {x \in S \iff \map P x}$

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

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