### Proper subset definition

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.