> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/200-set-theory/topics/membership.md).

# 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: $$x\notin S = \lnot(x\in S)$$