Set comprehension

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.

Defining 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 examples:

There is no order among the elements of a set (this explains and validates the equality of the last example), but with the ellipses notation we show an ordered sequence before (or after) the ellipsis as a convenient notational vehicle for explaining to a reader which elements are in a set. The first few elements of the sequence are shown then the ellipses indicate that the simplest interpretation should be applied for continuing the sequence. Should no terminating value appear to the right of the ellipses then the sequence is considered to be unbounded.

In each preceding example, each set is described by enumerating its elements. Not all sets can be described in this way, or if they can, their enumeration may be too long or too complicated to be useful. Therefore, many sets are defined by a property that characterizes their elements. This characterization may be done informally using general prose, as in the following example.

However, the prose approach may lack accuracy or be ambiguous. Thus, set builder notation is often used with a predicate characterizing the elements of the set being defined, as described in the following section.

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