Definition:Smallest Set by Set Inclusion/Class Theory
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Definition
Let $A$ be a class.
Then a set $m$ is the smallest element of $A$ (with respect to the subset relation) if and only if:
- $(1): \quad m \in A$
- $(2): \quad \forall S: \paren {S \in A \implies m \subseteq S}$
Also known as
The smallest element in this context is also referred to as the least element.
Also see
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 4$ A double induction principle and its applications: Definition $4.8$