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) 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 $\sqrt{a}+\frac{b}{c}=\frac{d}{e}$ where $\sqrt{a}$ is irrational and $b,c,d,e$ are integers
$\frac{d}{e}-\frac{b}{c}$ is a rational number and $\sqrt{a}$ is 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 $n^2>9$
Let us assume that $n$ is a real number with $n>3$ but $n^2>9$ is not true.
That is $n^2<9$
Then $n>3$ and $n$ is a real number
Squaring both the sides we obtain,
$n^2>{\left(3\right)}^2$
$\Rightarrow n^2>9$ which is a contradiction since we have assumed that $n^2<9$
Thus the given statement is true. That is if $n$ is a real number with $n>3$ then $n^2>9$