Elementary Properties of the Maximum of Two Elements

Let $S$ be a set equipped with a total order $\le$, let $a,b,c\in S$, and let $\max\{a,b\}$ denote the maximum of $a$ and $b$. Then the following hold.

1. (Upper bound) $a\le\max\{a,b\}$ and $b\le\max\{a,b\}$.

2. (Attainment) $\max\{a,b\}=a$ or $\max\{a,b\}=b$.

3. (Least upper bound) $\max\{a,b\}\le c$ if and only if both $a\le c$ and $b\le c$.

4. (Symmetry) $\max\{a,b\}=\max\{b,a\}$.