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 is a real number with then (by contradiction method)
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Text solutionVerified
(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 is false. That is we assume that the sum of an irrational number and a rational number is rational.
Therefore where is irrational and are integers
is a rational number and 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 is as follows,
If is a real number with then
Let us assume that is a real number with but is not true.
That is
Then and is a real number
Squaring both the sides we obtain,
which is a contradiction since we have assumed that
Thus the given statement is true. That is if is a real number with then
p : the sum of an irrational number and a rational number is irrational
Let us assume that the given statement is false. That is we assume that the sum of an irrational number and a rational number is rational.
Therefore where is irrational and are integers
is a rational number and 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 is as follows,
If is a real number with then
Let us assume that is a real number with but is not true.
That is
Then and is a real number
Squaring both the sides we obtain,
which is a contradiction since we have assumed that
Thus the given statement is true. That is if is a real number with then
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Question Text | 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 is a real number with then (by contradiction method) |
Answer Type | Text solution:1 |
Upvotes | 150 |