Definition:Proper Subset/Improper
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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:
- $S = \O$
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
- 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