If $(S, \leq)$ is a partial ordered class and $S$ is finite, how can I prove that there exists a minimal element in $S$? I kind of mix up the the terms "smallest" and "minimal" - that is why I dont really know what the approach is when showing that there exists a minimal element in a set.
Some help would be much appreciated.
An element $s \in S$ is minimal iff there is no smaller element, that is $$ \forall t \in S \quad t \le s \Rightarrow t = s $$ i. e. the only element smaller or equal to $s$ is $s$ itself.
$s$ is the smallest element, iff it is smaller then all elements, i. e. $$ \forall t \in S \quad s \le t. $$ (Note, that a smallest element is minimal, but not vice versa).
To show that a finite set has a minimal element, use e. g. induction on $|S|$: Pick $s \in S$, if $s$ is minimal, we are done, otherwise consider the non-empty set $\{t \in S : t < s\}$, which by induction has a minimal element. Now show that it is also a minimal element for $S$.