Question

Consider the following statement. If r is any rational number and s is any irrational number, then r s is irrational. (a) Which of the following is a negation for the statement? If r is any rational number and s is any irrational number, then r s is rational. If r is any irrational number and s is any rational number, then r s is irrational. There is a rational number r and an irrational number s such that r s is irrational. There is a rational number r and an irrational number s such that r s is rational. (b) What are some values of r and s for which the given statement is false (that is, for which its

Answer 1

Answer:

(a) If r is any rational number and s is any irrational number, then r/s is rational

(b) The statement is false when r is 0

Step-by-step explanation:

Given

$r \to$ rational number

$s \to$ irrational number

$\frac{r}{s} \to$ irrational number

Solving (a): The negation

To get the negation of a statement, we only need to negate the end result

In other words, the number type of r and s will remain the same, but r/s will be negated.

So, the negation is:

$r \to$ rational number

$s \to$ irrational number

$\frac{r}{s} \to$ rational number

Solving (b): When r/s is irrational is false

Given that:

$\frac{r}{s} \to$ irrational number

Set r to 0

So:

$\frac{r}{s} = \frac{0}{s}$

$\frac{r}{s} = 0$ -- rational

Hence, the statement is false when r is 0