Question

Prove the following statement using a proof by contraposition. Yr EQ,s ER, if s is irrational, then r + 1 is irrational.

Answer 1

Answer:

I think that what you are trying to show is:  If $s$ is irrational and $r$ is rational, then $r+s$  is rational. If so, a proof can be as follows:

Step-by-step explanation:

Suppose that $r+s$ is a rational number. Then $r$ and $r+s$ can be written as follows

$r=\frac{p_{1}}{q_{1}}, \,p_{1}\in \mathbb{Z}, q_{1}\in \mathbb{Z}, q_{1}\neq 0$

$r+s=\frac{p_{2}}{q_{2}}, \,p_{2}\in \mathbb{Z}, q_{2}\in \mathbb{Z}, q_{2}\neq 0$

Hence we have that

$r+s=\frac{p_{1}}{q_{1}}+s=\frac{p_{2}}{q_{2}}$

Then

$s=\frac{p_{2}}{q_{2}}-\frac{p_{1}}{q_{1}}=\frac{p_{2}q_{1}-p_{1}q_{2}}{q_{1}q_{2}}\in \mathbb{Q}$

This is a contradiction because we assumed that $s$ is an irrational number.

Then $r+s$ must be an irrational number.