Set-builder notation

Set-builder notation specifies a set by stating a property that its elements satisfy, often using a variable and a condition inside braces.

Set: A collection of distinct objects, treated as a single mathematical object. Set-builder notation is one way to specify which objects belong to a set.
Set comprehension: A method of defining a set by selecting objects that satisfy a specified condition. It is the general construction that set-builder notation writes compactly.
Interval: A subset of the real numbers containing every number between any two of its elements. An interval such as {x ∈ ℝ : 2 < x ≤ 5} illustrates a condition-defined set.
Predicate logic: A formal system of logic that uses predicates, variables, and quantifiers. Set-builder expressions rely on the same predicate-and-variable structure.
Set membership: The relation stating that an object is an element of a set, written with the symbol ∈. A set-builder condition determines which objects satisfy membership.
Roster notation: A way to specify a set by listing its elements between braces. It names elements directly, unlike set-builder notation’s property-based definition.
Solution set: The set of values that make an equation, inequality, or system true. Set-builder notation states precisely which values solve a mathematical problem.
Naive set theory: An informal approach to sets that treats any well-described collection as a set. Unrestricted property-based definitions expose a foundational problem in this approach.
Predicate: A statement with one or more variables that becomes true or false when values are supplied. The property inside set-builder braces is commonly expressed as a predicate.
Set-builder form: A notation such as {x ∈ A : P(x)} that selects elements of A satisfying P. This standard form makes the candidate domain and membership condition explicit.

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