An element of a set S is an object that belongs to S, written x ∈ S. The membership relation ∈ is the primitive notion of set theory: all other concepts (subset, function, relation) are defined through it. If x does not belong to S, we write x ∉ S.
Elements may themselves be sets — in ZFC set theory, everything is a set, so the elements of {1, 2, 3} are themselves sets (0 = ∅, 1 = {∅}, 2 = {∅, {∅}}, and so on). The element–set distinction is one of role, not of kind: the same object can be an element of one set and a set containing other elements.
In a category, an element of an object A is a morphism from the terminal object 1 to A. This “generalized element” perspective extends the concept beyond sets: in a topos, elements of the subobject classifier Ω are truth values, and in the semiotic universe, the elements of the Heyting algebra H are semantic values ordered by entailment.