Many of the questions we might want to answer about a given set involve its relationship to potential elements or to other sets.

Summary if you have math anxiety

The most basic thing you can ask about a set \(S\) is whether some object \(o\) is a member of it.  If it is, our shorthand for that is \(o \in S\).  Based on that we can ask whether the elements of one set \(S_1\) are also elements of another set \(S_2\).  If they are, we say \(S_1\) is contained in \(S_2\), and write it as \(S_1 \subseteq S_2\).  If the two sets are each contained in the other, they're the same set and we say \(S_1 = S_2\).

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Membership: \(\in\) and \(\notin\)

Membership is a relationship between an object and a set. If an object \(x\) is a member of a set \(S\), we write that as:

If it's not a member, then we write:

Containment: \(\subset\), \(\subseteq\) and \(\not\subseteq\)

A set \(A\) is a subset of another set $$B$ if all of \(A\)'s members are also members of \(B\).  We might also say \(A\) is contained in \(B\).  We notate containment as:

We consider sets to be contained in themselves (to be subsets of themselves), hence the \(\subseteq\) symbol having the bar on the bottom to make it look a little more like an equals sign.  In situations where \(B\) has elements that \(A\) doesn't we say that \(A\) is a proper subset of \(B\) and notate it without the extra bar:

If \(A\) has elements that \(B\) doesn't, then it's not contained in \(B\), notated: \(A\not\subseteq B\)

Equality: \(=\) and \(\neq\)

As we mentioned before, two sets \(A\) and \(B\) are equal when they have the exact same elements.  We notate that with an equals sign, just like with anything else:

If \(A\subseteq B\) and also \(B \subseteq A\), then \(A=B\).

If \(A\) and \(B\) don't have identical, they're not equal: \(A\neq B\).

Disjoint sets

If two sets are no elements in common, then we way they are disjoint.  There's no special symbol for this relationship, although, as we will see in set operations, we can express this by saying \(A\cap B = \emptyset\).