Definition:Proper Subset

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 the set $$S$$.

If $$S \subseteq T$$ and $$S \ne T$$, then the notation $$S \subset T$$ and $$T \supset S$$ 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 $$\varnothing \subset S \subset T$$.

Notes:

Some authors do not require that $$S \ne \varnothing$$ for $$S$$ to be a proper subset of $$T$$.

The literature can be confusing. Many authors use $$\subset$$ for what we have defined $$\subseteq$$ for. If it is important with this usage to indicate that $$S$$ is a proper subset of $$T$$, the notation $$S \subsetneq T$$ or $$T \supsetneq S$$ can be used.