Definition:Proper Subset

Definition
If a set $$S$$ is a subset of another set $$T$$, that is, $$\left({S \subseteq T}\right)$$, and also:


 * $$\left({S \ne T}\right)$$
 * $$\left({S \ne \varnothing}\right)$$

then $$S$$ is referred to as a proper subset of $$T$$.

The set $$T$$ properly contains, or strictly contains, the set $$S$$.

If $$S \subseteq T$$ and $$S \ne T$$, then the notation $$S \subset T$$ is used.

If we wish to refer to a set which we specifically require not to be empty, we can denote it like this:
 * $$\varnothing \subset S$$

... and one which we want to specify as possibly being null, we write:
 * $$\varnothing \subseteq S$$

Thus for $$S$$ to be a proper subset of $$T$$, we can write it as $$\varnothing \subset S \subset T$$.

Proper Superset
In a similar vein to the concept of a superset, $$T \supset S$$ means $$T$$ is a proper superset of $$S$$. This can be interpreted as $$T$$ properly contains $$S$$.