Definition:Non-Empty Set
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Definition
Let $S$ be a set.
Then $S$ is said to be non-empty if and only if $S$ has at least one element.
By the Axiom of Extension, this may also be phrased as:
- $S \ne \O$
where $\O$ denotes the empty set.
Many mathematical theorems and definitions require sets to be non-empty in order to avoid erratic results and inconsistencies.
Non-Empty Class
In the context of class theory, the definition follows the same lines:
Let $A$ be a class.
Then $A$ is non-empty if and only if $A$ has at least one element.
Also known as
Some sources prefer to use nonempty, but $\mathsf{Pr} \infty \mathsf{fWiki}$, in striving for consistency, standardises on non-empty.
Also see
Sources
- 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Introduction: $1.3$. Notation
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems