Definition:Proper Subset

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Definition

Let $S$ and $T$ be sets such that $S$ is a subset of $T$.

Let $S \ne T$.

Then $S$ is referred to as a proper subset of $T$, and we write:

$S \subsetneq T$

or:

$S \subsetneqq T$


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 $T \supsetneqq S$.


Also defined as

Some authors require that $S \ne \O$ for $S$ to be a proper subset of $T$.

Hence, under this convention, $S$ is a proper subset of $T$ if and only if

$\O \subsetneqq S \subsetneqq T$

It is wise to be aware of which definition is in use.


Improper Subset

The term improper subset is relevant in treatments of set theory which define a proper subset $T$ as a subset $S$ of $T$ such that:

$0 \subsetneqq S \subsetneqq T$


Under such a regime, $S$ is an improper subset of $T$ if and only if either:

$S = T$

or:

$S = \O$


Also known as

$S \subsetneqq T$ can also be read as:

$S$ is properly included in $T$, or $T$ properly includes $S$
$S$ is strictly included in $T$, or $T$ strictly includes $S$


The following usage can also be seen for $S \subsetneqq T$:

$S$ is properly contained in $T$, or $T$ properly contains $S$
$S$ is strictly contained in $T$, or $T$ strictly contains $S$


However, beware of the usage of contains: $S$ contains $T$ can also be interpreted as $S$ is an element of $T$.

Hence the use of contains is not supported on $\mathsf{Pr} \infty \mathsf{fWiki}$.


Notation

Notation in the literature for the concept of a subset can be confusing.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, the convention is as follows:

$\subseteq$ is used for the general subset
$\subsetneq$ or $\subsetneqq$ is used for the concept of a proper subset.

The notation $\subset$ is deprecated on $\mathsf{Pr} \infty \mathsf{fWiki}$, on the grounds that it can mean either.


Sources

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