Definition:Proper Subset/Improper

Definition

Let $T$ be a set.

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:

$S = \O$

Let us consider treatments of set theory which categorise the empty set $\O$ as a proper subset.

Then an improper subset $S$ of a set $T$ is such that:

$S = T$

As this is merely a matter of nomenclature, this distinction should not be of great importance.

However, it is wise to make sure which usage is intended when it is encountered.