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)

Answer 1

(i) Let us assume that p is not true.

$\therefore$ Sum of an irrational and a rational number is not irrational.

$\Rightarrow$ There exists an irrational number a and a rational number b such that a + b is not irrational.

$\Rightarrow a+b=c$ (say) is a rational number.

$\Rightarrow a=c-b$

$\Rightarrow$ a is rational.

But a is irrational, which is contradiction So our supposition in wrong.

Thus p is true.

(ii) Let r and s be the statements given by r : n is a real number with n > 3

$s:n^2>9$

Then, $q=r\Rightarrow s$

If possible let q is not true then

$\Rightarrow$ ~q is true

$\Rightarrow r$ and ~s is true.

$\Rightarrow n$ is a real number with n > 3 and $n^2<9$ which is contradiction

So our supposition is wrong.

Thus q is true.