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).
1 Answer
Solution :-
(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 statement p is false. That is,
The sum of an irrational number and a rational number is rational.
Hence,
where
\( \sqrt{a}\) is irrational and b, c, d, e are integers.
\( \dfrac{d}{e}-\dfrac{b}{c}=\sqrt{a}\)
But here, \( \dfrac{d}{e}-\dfrac{b}{c}\) is a rational number and \( \sqrt{a}\) is an irrational number
This is a contradiction. Hence, our assumption is false.
The sum of an irrational number and a rational number is rational.
Hence, 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
i.e. n2 < 9
So, n > 3 and n is a real number
By squaring both sides, we get
n2 > (3)2
This implies that n2 > 9 which is a contradiction, since we have assumed that n2 < 9
Therefore, the given statement is true i.e., if n is a real number with n > 3, then n2 > 9.
Answered by Abhisek | 1 year agoRelated Questions
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