Prove the following statement by contradiction method. p: The sum of an irrational number and a rational number is irrational

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Prove the following statement by contradiction method. p: The sum of an irrational number and a rational number is irrational

Solution:

Let’s say that $p$ is false, as sum of an irrational no. and a rational no. is irrational.

Let $\sqrt{\lambda }$ is irrational and $n$ is rational no.

$\sqrt{\lambda } + n = r$

$\sqrt{\lambda } = r - n$

But, it is known that $\sqrt{\lambda }$ is irrational whereas $(r-n)$ is rational which is a contradiction.

So here, our assumption is False.

As a result, $P$ is true.

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