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 n 2 9 (by contradiction method).

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 n 2 > 9 (by contradiction method).

Philoid · Accepted Answer

(i) Assume that the given statement p is false.

So, the statement becomes the sum of an irrational number and a rational number is rational.

Let us take for example,

Where √p is irrational number and q/r and s/t are rational numbers.

Then, is a rational number and √p is an irrational number.

This is a contradiction.

∴The assumption we made is wrong.

Thus, the given statement p is true.

(ii) Assume that the given statement q is false.

So, the statement becomes if n is a real number with n > 3, then n² < 9.

From the given statement, we know that n > 3 and n is a real number.

Squaring on both sides, we get

⇒ n² > 3²

⇒ n² > 9

This is a contradiction.

∴The assumption we made is wrong.

Thus, the given statement q is true.

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).

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).