Membership relation

The relation between an object and a set that holds exactly when the object is an element of that set. It is written ∈, with its negation written ∉.

Set: A collection treated as a single mathematical object, whose elements are specified by a defining condition or construction. Membership states whether an object belongs to one of these collections.
Set-builder notation: A notation that specifies a set by stating the condition its elements satisfy. It defines sets through membership conditions such as x ∈ A exactly when a predicate holds.
Subset relation: A relation between sets in which every element of one set belongs to the other. Unlike membership, it compares sets rather than an object with a set.
Russell's paradox: A contradiction arising from the set of all sets that do not contain themselves. Unrestricted membership-based set formation permits this self-reference paradox.
Element (mathematics): An object that belongs to a set. The relation connects a set to candidate elements.
Characteristic function: A function that assigns 1 to objects in a specified set and 0 to objects outside it. It encodes membership as a binary-valued function when a surrounding domain is fixed.
Set equality: The relation holding between two sets exactly when they have the same elements. Equality compares sets as wholes; membership asks whether an object belongs to one set.
Well-founded relation: A relation is well-founded when every nonempty subset of its domain has a minimal element. The axiom of foundation requires membership to be well-founded in standard Zermelo-Fraenkel set theory.
Equality: A relation asserting that two expressions denote the same object. In standard set theories, membership is unchanged when an element is replaced by an equal object.
Subset: A set A is a subset of B when every element of A is also an element of B. Its definition universally quantifies over membership in both sets.

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