Set-builder notation

${\displaystyle \{n\in \mathbb {Z} \mid (\exists k\in \mathbb {Z} )[n=2k]\}}$

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.

Defined sets by properties is also known as set comprehension, set abstraction or as defining a set's intension.

Set-builder notation is sometimes simply referred to as set notation, although this phrase may be better reserved for the broader class of means of denoting sets.

A set is an unordered collection of elements. (An element may also be referred to as a member). An element may be any mathematical entity.

A set can be described directly by enumerating all of its elements between curly brackets, as in the following two examples:

This is sometimes called the "roster method" for specifying a set.

When it is desired to denote a set that contains elements from a regular sequence an ellipses notation may be employed, as shown in the next two examples:

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