Definition:Improper Subset/Also defined as

Improper Subset: Also defined as

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.

Sources

• 1971: Patrick J. Murphy and Albert F. Kempf: The New Mathematics Made Simple (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets: Subsets
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.2$: Sets