Definition:Proper Subset

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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 empty, we write:

$\O \subseteq S$


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


Proper Superset

If $S$ is a proper subset of $T$, then $T$ is a proper superset of $S$.

This can be expressed by the notation $T \supsetneqq S$.


This can be interpreted as $T$ properly contains $S$.


Improper Subset

$S$ is an improper subset of $T$ if and only if $S$ is a subset of $T$ but specifically not a proper subset of $T$.

That is, either:

$S = T$

or:

$S = \varnothing$


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 $\mathsf{Pr} \infty \mathsf{fWiki}$ 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 $\subsetneqq$.


Sources