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).
(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,
,
where
is
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.