An element of a set S is an object that belongs to S, written $x \in S$. The membership relation $\in$ is the primitive notion of set theory: all other concepts — subset, function, binary relation — are defined through it. If x does not belong to S, we write $x \notin S$.

Membership is a yes-or-no question. For any object x and any set S, either $x \in S$ or $x \notin S$. There is no partial membership, no degree of belonging. This definiteness is what makes a set a set — without it, the collection is not well-defined.

Elements may themselves be sets. In ZFC set theory, everything is a set, so the elements of {1, 2, 3} are themselves sets (0 = $\emptyset$, 1 = {$\emptyset$}, 2 = {$\emptyset$, {$\emptyset$}}). 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. The number 3 is an element of the set {1, 2, 3} and also a set containing three elements.

An ordered pair is built from elements. A binary relation is a set of ordered pairs. A function is a relation where each element of the domain appears in exactly one pair. The chain from element to function passes through ordered pair and relation — each step builds on membership.

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: instead of asking “does x belong to A?” you ask “is there an arrow from 1 to A that picks out x?” In Set, these are the same question. In other categories, the generalized version is the one that works.