Smallest Element/Examples/Finite Subsets of Natural Numbers

Examples of Minimal Elements

Let $\FF$ denote the set of finite subsets of the natural numbers $\N$.

Consider the ordered set $\struct {\FF, \subseteq}$.

Then $\struct {\FF, \subseteq}$ has a smallest element, and that is the empty set $\O$.

Proof

We have that $\O$ is a finite set.

By Empty Set is Subset of All Sets it follows that $\O$ is a subset of $\N$.

Hence $\O \in \FF$ by definition of $\FF$.

Let $A \in \FF$ be some finite subset of $\N$.

Then by Empty Set is Subset of All Sets:

$\O \subseteq A$

Hence $\O$ is the smallest element of $\struct {\FF, \subseteq}$ by definition.

$\blacksquare$

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 $6 \ \text {(a)}$