Question

Use Exercise 25 to providea proof different from that in the text that the set of rational numbers is countable. [Hint: Show that you can express a rational number as a string of digits with a slash and possibly a minus sign].

Short answer

We will use the fact that every rational number is of the form $\frac{a}{b}$where a and b are integers and $b\ne 0$

Step-by-step solution

Step 1:

We will use the fact that every rational number is of the form $\frac{a}{b}$where a and b are integers.

Take the list of characters to be $\{0,1,2,3,4,5,6,7,8,9,/,-\}$and write every integer in the standard numeral system.

Then every rational number is labeled with a finite string of characters from our list of characters (Put the character to distinguish the two integers and the - character if the number is negative). Then we have satisfied the conditions of Exercise 25 and hence we have the result.