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Concept# Set-builder notation

Summary

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 (mathematics)#Roster notation
A set can be described directly by enumerating all of its elements between curly brackets, as in the following two examples:
is the set containing the four numbers 3, 7, 15, and 31, and nothing else.
is the set containing a, b, and c, and nothing else (there is no order among the elements of a set).
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 ellipsis notation may be employed, as shown in the next examples:
is the set of integers between 1 and 100 inclusive.
is the set of natural numbers.
is the set of all integers.
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 use an ordered sequence before (or after) the ellipsis as a convenient notational vehicle for explaining 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 general, denotes the set of all natural numbers such that . Another notation for is the bracket notation . A subtle special case is , in which is equal to the empty set . Similarly, denotes the set of all for .
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.

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