The Maximum and Minimum of Two Elements of a Total Order

In the setting of Sets and Maps: Ordinary Notation, let $\le$ be a total order on a set $X$, and let $a,b,c\in X$.

¶ The subset $\{a,b\}$ of $X$ has a greatest element $\max\{a,b\}$ and a least element $\min\{a,b\}$.

¶ If $a\le b$, then $\max\{a,b\}=b$ and $\min\{a,b\}=a$; if $b\le a$, then $\max\{a,b\}=a$ and $\min\{a,b\}=b$.

¶ $\min\{a,b\}\le a\le\max\{a,b\}$ and $\min\{a,b\}\le b\le\max\{a,b\}$.

¶ $\max\{a,b\}\le c$ if and only if $a\le c$ and $b\le c$.

¶ $c\le\min\{a,b\}$ if and only if $c\le a$ and $c\le b$.

¶ $c<\max\{a,b\}$ if and only if $c<a$ or $c<b$; and $\min\{a,b\}<c$ if and only if $a<c$ or $b<c$.