Definition:Proper Subset

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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 $S$.


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 $\subseteq$ to be. 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.


Sources

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