Smallest Element/Examples/Finite Subsets of Natural Numbers less Empty Set
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Examples of Smallest Elements
Let $\FF$ denote the set of finite subsets of the natural numbers $\N$.
Let $\GG$ denote the set $\FF \setminus \set \O$, that is, $\FF$ with the empty set excluded.
Consider the ordered set $\struct {\GG, \subseteq}$.
$\struct {\GG, \subseteq}$ has no smallest element.
Proof
From Minimal Element: Finite Subsets of Natural Numbers less Empty Set, $\struct {\GG, \subseteq}$ has more than one minimal element.
The result follows from Ordered Set with Multiple Minimal Elements has no Smallest Element.
$\blacksquare$
Sources
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations: Exercise $6 \ \text {(a)}$