setbuilder

Setbuilder, in mathematics, refers to a notation used to describe a set by listing a defining property or predicate that its elements must satisfy. The standard form is { x ∈ A | P(x) }, meaning the set of all elements x in A for which the property P(x) holds. Other common forms use a colon instead of the vertical bar, such as { x ∈ A : P(x) }, or a general form like { f(x) : x ∈ A, P(x) }. The idea is to specify a set indirectly rather than by enumerating its members.

In practice, setbuilder notation is used to describe sets concisely and flexibly. It can define subsets of a given universe, construct sets by selecting elements that meet a condition, or describe images and preimages under mappings. Examples include { n ∈ N | n is even } for even natural numbers, { x^2 : x ∈ Z } for perfect squares, and { x ∈ R | x^2 < 2 } for real numbers with square less than two. The notation is also adapted to more complex expressions, such as { f(x) : x ∈ X, P(x) } or { (x, y) ∈ X × Y | P(x, y) }.

Historically, setbuilder notation became standard as set theory and formal logic developed, and it is closely tied to the development of axiomatic systems that replace unrestricted comprehension with restricted forms. In modern mathematics, its use is widespread in proofs, definitions, and formal specifications, and it has parallels in computer science through constructs like list comprehensions.