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:
- {\{x \mid \Phi(x)\}} or
- {\{x : \Phi(x)\}}
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 \emptyset or \{ \}}.