Subsets

Recall that a set is a collection of elements.

A set $A$ is a subset of a set $B$ if every element in $A$ is also in $B$ .

For example, if $A=\{1,3,5\}$ and $B=\{1,2,3,4,5\}$ , then $A$ is a subset of $B$ , and we write

$A\subseteq B$

The line under the sideways $\cup$ means that $A$ may also be equal to $B$ (that is, they may be identical sets). If we want to say that $A$ is a proper subset of $B$ (that means: it's a subset, but there is at least one element in $B$ that is not in $A$ ) then we can remove the line:

$A\subset B$

To write that a set is not a subset of another set, just put a slash through the sideways $\cup$ :

$B\not\subset A$