Set-builder notation

Set-builder notation is a mathematical notation used in set theory, logic, mathematics, and computer science to describe a set by stating the properties that its members must satisfy. Defining sets by properties is also known as set comprehension, set abstraction, or defining a set's intension.

 Structure 
In set-builder notation, a set is typically written in one of the following forms:
  or 


Where:
 x is a variable representing an element of the set.
 The vertical bar "∣" (or sometimes a colon ":") is interpreted as "such that" or "for which."
 Φ(x) is a predicate (a logical formula) that must be satisfied by the elements of the set.

 Use in Predicate Logic 
Set-builder notation is heavily used in predicate logic to define sets based on conditions. If the predicate (the condition) is true for a particular element, that element belongs to the set.

 Empty Sets 
If no values satisfy the condition specified in set-builder notation, the set will be the empty set, denoted by  or .

 See also 
 Set (mathematics)
 Set theory
 Predicate logic
 Naive set theory
 List of mathematical symbols

 References 

Category:Mathematical notation