A proper subset of a set $A$ is a subset of $A$ that is not equal to $A$. In other words, if $B$ is a proper subset of $A$, then all elements of $B$ are in $A$ but $A$ contains at least one element that is not in $B$.

For example, if $A =\{1,3,5\}$ then $B=\{1,5\}$ is a proper subset of $A$. The set $C=\{1,3,5\}$ is a subset of $A$, but it is not a proper subset of $A$ since $C=A$. The set $D=\{1,4\}$ is not even a subset of $A$, since 4 is not an element of $A$.

See also proper superset.