Definition:Proper Subset/Also known as
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Proper Subset: Also known as
$S \subsetneqq T$ can also be read as:
- $S$ is properly included in $T$, or $T$ properly includes $S$
- $S$ is strictly included in $T$, or $T$ strictly includes $S$
The following usage can also be seen for $S \subsetneqq T$:
- $S$ is properly contained in $T$, or $T$ properly contains $S$
- $S$ is strictly contained in $T$, or $T$ strictly contains $S$
However, beware of the usage of contains: $S$ contains $T$ can also be interpreted as $S$ is an element of $T$.
Hence the use of contains is not supported on $\mathsf{Pr} \infty \mathsf{fWiki}$.
Sources
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 1$: Operations on Sets
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Subsets and Complements; Union and Intersection
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 1$: The Axiom of Extension
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 17$: Finite Sets
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.3$: Subsets: Definition $3.2$
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 6$: Subsets
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): inclusion
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): proper inclusion
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): proper inclusion
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): properly included