Chapter 5, Problem 1E

Explanation of Solution

Definition of natural numbers with leading zeros:

From definition of natural number given in the text book, the natural number “n” can be,

$n0\left|n1\right|n2\left|\mathrm{...}\right|n9$

Where, “n” can be any numbers $0\left|1\right|\mathrm{...}|9$

Therefore, natural number can be a number with any number of leading zeros.

To get natural number with non leading zeros, modify the definition as following.

Definition of natural numbers with non leading zeros:

$\begin{array}{l}\text{n::=0|1|}\mathrm{...}\text{|9|1m|}\mathrm{...}\text{|9m}\\ \text{m::=0|1|}\mathrm{...}\text{|9|0m|1m|}\mathrm{...}\text{|9m}\end{array}$

Number “n” can be:

$\text{0|1|}\mathrm{...}\text{|9}$

Or,

$1m\left|2m\right|\mathrm{...}|9m$

• Where, “m” can be any natural number. Therefore, “n” can be “0” and all natural numbers with non leading zeros.
• This is because of the recursion definition of “n” does not staring with “0”. It is starting with $1,2,\dots 9.$