Definition:Set Definition by Predicate/Also known as

Set Definition by Predicate: 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.

Some sources call such a construction as a set former.

Some sources refer to it as definition by characteristic property.

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.

A common variant for presenting a conjunction of propositional functions:

$\set {x: \map P x \wedge \map Q x}$

is:

$\set {x: \map P x, \map Q x}$

Some sources use the notation:

$\boldsymbol [x: \map P x \boldsymbol ]$

for $\set {x: \map P x}$.