Check the validity of the statement given below by the method specified against it.
$p:$ The sum of an irrational number and a rational number is irrational (by contradiction method).
$p:$ The sum of an irrational number and a rational number is irrational (by contradiction method).
(A) The given statement is: $p:$ The sum of an irrational number and a rational number is irrational.
To check the validity by the contradiction method,we assume the negation of the statement is true.
Let us assume that the sum of an irrational number $x$ and a rational number $y$ is a rational number $z$.
So,$x + y = z$,where $x$ is irrational and $y, z$ are rational.
This implies $x = z - y$.
Since the difference of two rational numbers $(z - y)$ is always a rational number,this implies that $x$ is a rational number.
This contradicts our initial premise that $x$ is an irrational number.
Therefore,our assumption that the sum is rational must be false.
Hence,the sum of an irrational number and a rational number is irrational. The statement $p$ is true.
To check the validity by the contradiction method,we assume the negation of the statement is true.
Let us assume that the sum of an irrational number $x$ and a rational number $y$ is a rational number $z$.
So,$x + y = z$,where $x$ is irrational and $y, z$ are rational.
This implies $x = z - y$.
Since the difference of two rational numbers $(z - y)$ is always a rational number,this implies that $x$ is a rational number.
This contradicts our initial premise that $x$ is an irrational number.
Therefore,our assumption that the sum is rational must be false.
Hence,the sum of an irrational number and a rational number is irrational. The statement $p$ is true.
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