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 $P \left({x}\right)$.

Hence a set can be specified by means of such a propositional function, e.g.:

$S = \left\{{x: P \left({x}\right)}\right\}$

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 $P \left({x}\right)$ is true.

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

This is sometimes known as the set-builder notation.

Also known as

An alternative notation for this is $S = \left\{{x \mid P \left({x}\right)}\right\}$, 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 well be prudent to use $\mid$ for such that after all.

Sometimes, it is convenient to abbreviate the notation by simply writing $S = \left\{{P \left({x}\right)}\right\}$ or even just $S = \left\{{P}\right\}$.

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

Axiomatic Set Theory

In the context of axiomatic set theory, a more strictly rigorous presentation of this concept is:

$S = \left\{{x \in A: P \left({x}\right)}\right\}$

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 $P \left({x}\right)$ 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:

$S = \left\{{s: s \in S}\right\}$

See the Axiom of Specification.

Also see

• Explicit Set Definition
• Implicit Set Definition

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 2$: The Axiom of Specification
• Steven A. Gaal: Point Set Topology (1964)... (previous)... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 1.1$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 1$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 4$
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 1.1$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 3$