Let's consider a finite non-empty set $A \,=\, \{a_1,\,\ldots,\,a_n\}$ of real numbers or an infinite but compact (i.e. bounded and closed) set $A$ of real numbers. In both cases the set has a unique least number and a unique greatest number.

• The least number of the set is denoted by $\min\{a_1,\,\ldots,\,a_n\}$ or $\min{A}$ .
• The greatest number of the set is denoted by $\max\{a_1,\,\ldots,\,a_n\}$ or $\max{A}$ .

In both cases we have $$\min{A} \;=\; \inf{A},$$ $$\max{A} \;=\; \sup{A},$$ $$\min{A} \;\leqq\; x \;\leqq\; \max{A} \quad \forall x\in A,$$ where $\inf{A}$ and $\sup{A}$ are the infimum and supremum of the set $A$ .

The $\min$ and $\max$ are set functions, i.e. they map subsets of a certain set to $\mathbb{R}$ .

The minimal and maximal number of a set of two real numbers obey the formulae $$\min\{a,\,b\} \;=\; \frac{a\!+\!b}{2}\!-\!\frac{|a\!-\!b|}{2},$$ $$\max\{a,\,b\} \;=\; \frac{a\!+\!b}{2}\!+\!\frac{|a\!-\!b|}{2},$$ $$\max\{a,\,-a\} \;=\; |a|$$