Subset, superset

When all members of set A are present in another set B, then A is a subset of B. Let’s say set B is the set of all movies ever produced. Then A (movies I’ve watched) is clearly a subset of B. This notion is expressed like so:

A \subseteq B

To look at things from the other end, B is a superset of A:

B \supseteq A

You know what else is a subset of B? An empty set!

\varnothing \subseteq B

This either sounds absolutely natural to you or extremely weird. It makes perfect sense to a mathematician, because it’s easy to argue: all members of \varnothing are present in B, all zero of them. It gets weirder. As per our definition, if all members of a set are also present in another set, then the one is a subset of the other. This means any set is a subset of itself.

A \subseteq A B \subseteq B Z \subseteq Z

By extension, if two sets are the same, then either of them is a subset of the other.

\textrm{if } A \subseteq B \textrm{ and } B \subseteq A \textrm{, then } A = B

When we look at a statement A \subseteq B, we often need to know whether A = B or not. To distinguish between the two cases, mathematicians use a special notion of a proper subset. If A \subseteq B, but A \neq B, then A is a proper subset of B.

A \subset B

Since I haven’t watched all the movies ever produced, I can say that A is a proper subset of B. So, a set is a subset of itself, but is never a proper subset of itself.