Question

Show that the set of real numbers that are solutions of quadratic equations $a{x}^2+bx+c=0$, where a, b and c are integers, is countable.

Short answer

The set of all real numbers that are solutions to the quadratic equations $a{x}^2+bx+c=0$with a, b, c integers is countable.

Step-by-step solution

Step 1:

DEFINITIONS

A set is countable if it is finite or countably infinite.

A set is finite if it contains a limited number of elements.

A set is countably infinite if the set contains an unlimited number of elements and if there is a one-to-one correspondence with the positive integers.

Step 2:

$a{x}^2+bx+c=0$To proof: The set of all real numbers that are solutions to the quadratic equations with a, b, c integers is countable.

PROOF:

The set of all integers is countable, as we can list 0 first and then list the positive and negative integers alternately in increasing order (of absolute value).

However, the set of all 3-tuples (a, b, c) with a, b, c integers is then countable as well (because $Z^{+}\times Z^{+}$was countable followed from being countable (proven in one of the previous exercises) and thus is countable will follow from Z being countable in a similar manner).

Since any quadratic equation $a{x}^2+bx+c=0$has at most two roots, we can then list all roots in the same order as the -tuples (a, b, c) and thus the corresponding set is then countable. However, this set is then the set of all real numbers that are solutions to the quadratic equations $a{x}^2+bx+c=0$with a, b, c integers and thus the set of all real numbers that are solution to the quadratic equations $a{x}^2+bx+c=0$with a, b, c integers is countable.