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 $T \supsetneqq S$.


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.


Examples

Arbitrary Example

Let $A = \set {1, 2, 3}$.

Let $B = \set {1, 2, 3, 4}$.

Let $C = \set {1, 2, 3}$.


We see by inspection that $A$ is a proper subset of $B$.

However, while $A$ is a subset of $C$, it is not the case that $A$ is a proper subset of $C$.


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.


Also denoted as

The symbology used by some authors to denote a proper subset is:

$A \subset B$

to mean:

$A$ is a proper subset of $B$.

While this practice is common, it is not recommended on $\mathsf{Pr} \infty \mathsf{fWiki}$ because it is also used by many other authors to mean subset which may or may not be proper.


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


Also see

  • Results about proper subsets can be found here.


Sources

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