Minimum

Let $(X,\le)$ be an ordered set and let $S\subseteq X$. An element $m\in S$ is a minimum of $S$ (written $m=\min S$) if

A minimum is a lower bound that actually lies in the set. If a minimum exists, it is unique and equals the infimum : $\min S=\inf S$.

Examples:

• In $\mathbb{R}$, $\min[0,1]=0$.
• In $\mathbb{R}$, the set $(0,1)$ has no minimum (but it has $\inf(0,1)=0$).
• In $\mathbb{Z}$, $\min\{n\in\mathbb{Z}: n\ge 3\}=3$.