Minimum

Last updated Nov 1, 2022

Let $(P, \leq)$ be a Partial Ordering. Then a Minimum of $A \subset P$ is any $x \in A$ s.t. $\forall y \in A$, $x \leq y$.

1. The minimum may not exist. A simple example is the set $\mathbb{R}$. For every $x \in \mathbb{R}$, $x \not\leq x-1$. Thus there is no Minimum in $\mathbb{R}$.
2. If it exists, A Minimum in a Total Ordering is Unique