Question

Check the validity of the statements given below by the method given against it.

(i) p: The sum of an irrational number and a rational number is irrational (by contradiction method).

(ii) q: If n is a real number with n > 3, then n² > 9 (by contradiction method).

Answer 1

(i) The given statement is as follows.
p: the sum of an irrational number and a rational number is irrational.

Let us assume that the given statement, p, is false. That is, we assume that the sum of an irrational number and a rational number is rational.

Therefore, , whereis irrational and b, c, d, e are integers.

is a rational number andis an irrational number.

This is a contradiction. Therefore, our assumption is wrong.

Therefore, the sum of an irrational number and a rational number is rational.

Thus, the given statement is true.

(ii) The given statement, q, is as follows.

If n is a real number with n > 3, then n² > 9.

Let us assume that n is a real number with n > 3, but n² > 9 is not true.

That is, n² < 9

Then, n > 3 and n is a real number.

Squaring both the sides, we obtain

n² > (3)²

⇒ n² > 9, which is a contradiction, since we have assumed that n² < 9.

Thus, the given statement is true. That is, if n is a real number with n > 3, then n² > 9.