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

For example, if $A =\{1,3,5\}$ then $B=\{1,3,4,5\}$ is a proper superset of $A$. The set $C=\{1,3,5\}$ is a superset of $A$, but it is not a proper superset of $A$ since $C=A$. The set $D=\{1,3, 7\}$ is not even a superset of $A$, since $D$ does not contain the element 5.

See also proper subset.