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.[1][2]

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 \{ \.

See also

References

  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.