Element of a set

An element of a set is an object that belongs to that set. Membership is written with the symbol ∈, as in x ∈ A.

Set: A collection of distinct objects, treated as a single mathematical object. An element is defined by its membership in a set.
Extensionality: The principle that sets with exactly the same elements are equal. It makes membership determine a set's identity.
Subset: A set A is a subset of B when every element of A is also an element of B. Unlike elementhood, subsethood relates one set to another set.
Power set: The set of all subsets of a given set. Its elements are sets, illustrating that sets can themselves be members.
Set membership: The relation stating that an object belongs to a set, written ∈. This relation is exactly what elementhood asserts.
Axiom of pairing: An axiom asserting that any two objects belong together to some set. It guarantees sets can be formed with specified elements.
Proper subset: A subset that is not equal to the set containing it. It distinguishes containment between sets from membership of an object.
Cartesian product: The set of ordered pairs formed by taking one entry from each of two sets. Membership in a product is determined by the pair's entries belonging to its factors.
Set-builder notation: A notation that defines a set by specifying a property its elements satisfy. It describes elements through the conditions that determine membership.
Axiom of union: An axiom asserting that the union of a set of sets is itself a set. Membership in a union reduces to membership in one of its component sets.