Definition:Proper Subset

Definition
If a set $S$ is a subset of another set $T$, that is, $S \subseteq T$, and also:


 * $S \ne T$
 * $S \ne \O$

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 $S \subsetneqq T$ is used.

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

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

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

Also defined as
Some authors do not require that $S \ne \O$ for $S$ to be a proper subset of $T$.

The symbol $\subsetneq$ is the usual form to use, but the ugly and unwieldy $\subsetneqq$ is used on after complaints that $\subsetneq$ is too subtle.

The literature can be confusing. Many authors use $\subset$ for what we have defined $\subseteq$ to be. Others use $\subset$ to mean $\subsetneqq$.

Because of this confusion, this website does not endorse the use of $\subset$, however neater it is than the unaesthetic and clumsy $\subsetneqq$.