Set membership

Sets and propositions

A proposition has a truth value - it is either true or not true - so it may be used with logical connectives such as not, and, or. Unlike propositions, sets or its elements don't have a truth value per se, so in order to use sets with logical connectives, there has to be a relation between a set and its constituting objects. This is where the membership relation comes in.

Membership relation

Membership relation, also called belongs-to or "is-an-element-of, is the fundamental binary relation between an object and a set, denoted using a variant of the Greek letter epsilon, ε.

A set $S$ is a collection of distinct objects, each of which is called an element (or member) of $S$. For an object $x$, we denote by $x \in S$ that an object $x$ is indeed a member of $S$; otherwise we write $x \notin S$.

The proposition $x \in S$, rarely $S \ni x$, is read as an object x belongs to a set $S$, or an object $x$ is an element of a set $S$.

When we substitute concrete objects for these variables, if we descover that the $x$ really does belong to the set $S$, then we say that the membership relation holds, i.e. that the proposition $x \in S$ is true.

If we descover that the object $x$ does not belong to the set $S$, then the proposition $x \in S$ is false, and its negation is true: $x \notin S$.

Non-membership relation may be defined in terms of the membership relation, as negated membership relation: $x\notin S = \lnot(x\in S)$