Question

Prove or disprove: a) If r and s are both rational numbers, then r+s is rational. b) If r+s is rational, then r and s are both rational.

Answer 1

If r+s is rational, then r and s are both rational.

Step-by-step explanation:

Statement: If r+s is rational, then r and s are both rational.

To prove: If r+s is rational, then r and s are both rational.

Proof:

1. Assume that r+s is rational.
2. By definition, a number is rational if it can be expressed as the ratio of two integers.
3. Let r = a/b and s = c/d, where a, b, c, and d are integers and b and d are not zero.
4. Therefore, r+s = (a/b) + (c/d) = (ad+bc)/bd.
5. Since ad+bc and bd are both integers, r+s is rational.
6. Therefore, if r+s is rational, then r and s are both rational.

Thus, we have proven that if r+s is rational, then r and s are both rational.