Definition:Proper Subset/Improper

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Let $T$ be a set.

$S$ 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:

$S = T$


$S = \O$

Also defined as

Some sources categorise the empty set $\O$ as a proper subset, and not an improper subset.

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.