ISC (Arts) Class 11CISCE
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Here to check whether a statement p is true, we assume that p is not true i.e. ∼p is true. Then, we arrive at some result which contradicts our assumption. Therefore, we conclude that p is true.

Prove that : sqrt 7 is irrational .
In this method, we assume that the given statement is false.
That is we assume that  sqrt 7 is rational.
This means that there exists positive integers a and b such that sqrt 7 = a/b , where a and b have no common factors.
Squaring the equation , we get
7 = a^2 / b^2 => a^2 = 7b^2 => 7 divides a. Therefore, there exists an integer c such  that a = 7c.
Then  a^2 = 49c^2  and  a^2 = 7b^2 Hence, 7b^2 = 49c^2 ⇒ b^2 = 7c^2 ⇒ 7 divides b.
But we have already shown that 7 divides a. This implies that 7 is a common factor of both of a and b which contradicts our earlier assumption that a and b have no common factors. This shows that the assumption sqrt 7 is rational is wrong. Hence, the statementsqrt 7 is irrational is true.