Maximum

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

A maximum is an upper bound that actually lies in the set. If a maximum exists, it is unique and equals the supremum : $\max S = \sup S$.

Examples:

• In $\mathbb{R}$, $\max[0,1]=1$.
• In $\mathbb{R}$, the set $(0,1)$ has no maximum (but it has $\sup(0,1)=1$).
• In $\mathbb{Z}$, $\max\{n\in\mathbb{Z}: n\le 5\}=5$.