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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.
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.2$. Sets
- 1971: Patrick J. Murphy and Albert F. Kempf: The New Mathematics Made Simple (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets: Subsets
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 2$: Sets and Subsets
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.2$: Sets