Elementary Properties of the Minimum of Two Elements

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

1. (Lower bound) $\min\{a,b\}\le a$ and $\min\{a,b\}\le b$.

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

3. (Greatest lower bound) $c\le\min\{a,b\}$ if and only if both $c\le a$ and $c\le b$.

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

5. (Comparison with the maximum) $\min\{a,b\}\le\max\{a,b\}$.