Greatest Element/Examples/Finite Subsets of Natural Numbers less Empty Set
Jump to navigation
Jump to search
Examples of Greatest Elements
Let $\FF$ denote the set of finite subsets of the natural numbers $\N$.
Let $\GG$ denote the set $\FF \setminus \O$, that is, $\FF$ with the empty set excluded.
Consider the ordered set $\struct {\GG, \subseteq}$.
$\struct {\FF, \subseteq}$ has no greatest element.
Proof
Aiming for a contradiction, suppose $A \in \GG$ is the greatest element of $\struct {\GG, \subseteq}$.
From Greatest Element is Maximal, $A$ is a maximal element of $\struct {\GG, \subseteq}$.
But from Maximal Element: Finite Subsets of Natural Numbers less Empty Set, $\struct {\GG, \subseteq}$ has no maximal element.
Hence $A$ cannot be the greatest element of $\struct {\GG, \subseteq}$.
The result follows from Proof by Contradiction.
$\blacksquare$
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 $6 \ \text {(a)}$