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 $\{ \$ .

References

↑ Rosen, Kenneth (2006-07-26). Discrete Mathematics and Its Applications. McGraw-Hill Education. ISBN 978-0-07-322972-0.
↑ "For Comprehensions". Scala Documentation. Retrieved 2024-10-07.

[1] Rosen, Kenneth (2006-07-26). Discrete Mathematics and Its Applications. McGraw-Hill Education. ISBN 978-0-07-322972-0.

[2] "For Comprehensions". Scala Documentation. Retrieved 2024-10-07.

Set-builder notation

Structure

Use in Predicate Logic

Empty Sets

See also

References