Let $x$ and $y$ be rational and irrational numbers,respectively. Is $x+y$ necessarily an irrational number? Give an example in support of your answer.

(A) Yes,$x+y$ is necessarily an irrational number.
Proof by contradiction:
Suppose $x+y = r$,where $r$ is a rational number.
Since $x$ is a rational number,we can write $y = r - x$.
Since the difference of two rational numbers ($r$ and $x$) is always a rational number,$y$ must be a rational number.
However,this contradicts the given information that $y$ is an irrational number.
Therefore,our assumption is wrong,and $x+y$ must be an irrational number.
Example:
Let $x = 5$ (rational) and $y = \sqrt{2}$ (irrational).
Then,$x+y = 5 + \sqrt{2} = 6.4142...$,which is a non-terminating and non-repeating decimal.
Hence,$x+y$ is an irrational number.