Question

If it is possible to label each element of an infinite set S with a finite string of keyboard characters, from a finite list characters, where no two elements of Shave the same label, then S is a countably infinite set. Use the above statement and prove that the set of rational numbers is countable.

A. We can label the rational numbers with strings from the set (0, 1, 2, 3, 4, 5, 6, 7, 8, 9,-) by writing down the string that represents that rational number in its simplest form. As the labels are unique, it follows that the set of rational numbers is countable.
B. We can label the rational numbers with strings from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9,/,-) by writing down the string that represents that rational number in its simplest form. As the labels are unique, it follows that the set of rational numbers is countable.
C. We can label the rational numbers with strings from the set (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1) by writing down the string that represents that rational number in its simplest form. As the labels are unique, it follows that the set of rational numbers is countable.
D. We can label the rational numbers with strings from the set (1, 2, 3, 4, 5, 6, 7, 8, 9,/ -) by writing down the string that represents that rational number in its simplest form. As the labels are unique, it follows that the set of rational numbers is countable.

Answer 1

D. We can label the rational numbers with strings from the set (1, 2, 3, 4, 5, 6, 7, 8, 9, / -) by writing down the string that represents that rational number in its simplest form. As the labels are unique, it follows that the set of rational numbers is countable.

Explanation:

The label numbers are rational if they are integers. The whole number subset is rational which is written by the string. The sets of numbers are represented in its simplest forms. The rational numbers then forms numbers sets which are countable.