Maximal Element/Examples/Finite Subsets of Natural Numbers less Empty Set
Examples of Minimal 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 $\struct {\GG, \subseteq}$.
There are no maximal elements of $\struct {\GG, \subseteq}$.
Proof
Aiming for a contradiction, suppose $A \in \GG$ is a maximal element of $\struct {\GG, \subseteq}$.
Then $A$ consists of a finite number of natural numbers.
Let $M = \map \max A$ denote the maximum of the elements of $A$.
Now consider the set:
- $A' = A \cup \set {M + 1}$
We have that $A'$ is a finite subset of $\N$.
Hence $A' \in \GG$.
But also from Set is Subset of Union:
- $A \subseteq A'$
while because $M + 1 \notin A$:
- $A' \ne A$
Hence $A$ cannot be a maximal 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 $6 \ \text {(a)}$