minimal and maximal number

Let’s consider a finite non-empty set $A=\{a_1,\mathrm{\dots },a_n\}$   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 $\min \{a_1,\mathrm{\dots },a_n\}$  or  $\min A$.

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

In both cases we have

$$\min A=infA,$$

$$\max A=supA,$$

$$\min A\leqq x\leqq \max A\mathit{\hspace{1em}}\forall x\in A,$$

where  $infA$  and  $supA$  are the infimum and supremum of the set $A$.

The $\min$ and $\max$ are set functions, i.e. they mapsubsets of a certain set to $ℝ$.

The $\min$ and $\max$ have the following distributive propertieswith respect to addition:

$$\min \{a_1,\mathrm{\dots },a_n\}+b=\min \{a_1+b,\mathrm{\dots },a_n+b\}$$

$$\max \{a_1,\mathrm{\dots },a_n\}+b=\max \{a_1+b,\mathrm{\dots },a_n+b\}$$

The minimal and maximal number of a set of two real numbers obeythe formulae

$$\min \{a,b\}=\frac{a+b}{2}-\frac{|a-b|}{2},$$

$$\max \{a,b\}=\frac{a+b}{2}+\frac{|a-b|}{2},$$

$$\max \{a,b\}-\min \{a,b\}=|a-b|,$$

$$\max \{a,b\}+\min \{a,b\}=a+b,$$

$$\max \{a,-a\}=|a|$$