Question

Give a recursive definition of the functions max and min so that $\left(a_1,a_2,\dots .a_n\right)\mathbf{\text{and}}\mathbf{min}\left(a_1,a_2,\dots .a_n\right)$are the maximum and minimum of the n numbers ${\mathbf{a}}_{\mathbf{1}}\mathbf{,}{\mathbf{a}}_{\mathbf{2}}\mathbf{,}\mathbf{\dots }\mathbf{.}{\mathbf{a}}_{\mathbf{n}}$respectively.

Short answer

max $\left({\mathrm{a}}_1,{\mathrm{a}}_{2,\dots }{\mathrm{a}}_{\mathrm{n}},{\mathrm{a}}_{\mathrm{n}+1}\right)=max\left(max\left({\mathrm{a}}_1,{\mathrm{a}}_{2,\dots },{\mathrm{a}}_{\mathrm{n}}\right),{\mathrm{a}}_{\mathrm{n}+1}\right)$and

min$\left({\mathrm{a}}_1,{\mathrm{a}}_{2,\dots }{\mathrm{a}}_{\mathrm{n}},{\mathrm{a}}_{\mathrm{n}+1}\right)=min\left(min\left({\mathrm{a}}_1,{\mathrm{a}}_{2,\dots },{\mathrm{a}}_{\mathrm{n}}\right),{\mathrm{a}}_{\mathrm{n}+1}\right)$

Step-by-step solution

The recursive definition of the sequence:

The recursive sequence is a sequence of numbers indexed by an integer and generated by solving a recurrence equation.

To give a recursive definition of the functions max and min  

When n = 1 , the maximum and minimum is the number itself.

$$\begin{gathered} \max \left({\mathrm{a}}_1\right)={\mathrm{a}}_1 \\ \min \left({\mathrm{a}}_1\right)={\mathrm{a}}_1 \end{gathered}$$

When n = 2 , the maximum is defined as the larger value and minimum is defined as the smaller value.

$\begin{array}{r}max\left({\mathrm{a}}_1,{\mathrm{a}}_2\right)=\left\{\begin{array}{ll}{\mathrm{a}}_1& \text{if}{\mathrm{a}}_1\ge {\mathrm{a}}_2\\ {\mathrm{a}}_2& \text{otherwise}\end{array}\right.\\ min\left({\mathrm{a}}_1,{\mathrm{a}}_2\right)=\left\{\begin{array}{ll}{\mathrm{a}}_2& \text{if}{\mathrm{a}}_1\ge {\mathrm{a}}_2\\ {\mathrm{a}}_1& \text{otherwise}\end{array}\right.\end{array}$

When n > 2 , the maximum of the n numbers is equal to the maximum of the first n - 1 numbers and the nth number, while the minimum of the n numbers is equal to the minimum of the first n - 1 numbers and the nth number.

Thus,

max$\left({\mathrm{a}}_1,{\mathrm{a}}_{2,\dots }{\mathrm{a}}_{\mathrm{n}},{\mathrm{a}}_{\mathrm{n}+1}\right)=max\left(max\left({\mathrm{a}}_1,{\mathrm{a}}_{2,\dots },{\mathrm{a}}_{\mathrm{n}}\right),{\mathrm{a}}_{\mathrm{n}+1}\right)$and

$\left({\mathrm{a}}_1,{\mathrm{a}}_{2,\dots }{\mathrm{a}}_{\mathrm{n}},{\mathrm{a}}_{\mathrm{n}+1}\right)=min\left(min\left({\mathrm{a}}_1,{\mathrm{a}}_{2,\dots },{\mathrm{a}}_{\mathrm{n}}\right),{\mathrm{a}}_{\mathrm{n}+1}\right)$