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I believe the statement is false. Because in order for $x + y$ to be rational, both x and y must be rational, and in order for $x + y$ to be irrational, either $x$ or $y$ must be irrational. I'm just not sure how to prove it. I'm trying to prove the statement is false by proving its negation is true. Here's what I have so far:

The statement is false. Its negation is "for all real numbers $x$ and $y$, either $x - y$ is not rational or $x + y$ is not irrational". In other words, "for all real numbers $x$ and $y$, either $x - y$ is irrational or $x + y$ is rational". Assume x and y are real numbers.

I'm not sure if I'm right up to this point or where to go from here if I am.

Alessio K
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2 Answers2

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Here is an example: $a=\sqrt{2}+1$ and $b=\sqrt{2}$. Then $a-b=1$ is rational but $a+b=2\sqrt{2}+1$ is irrational (if it was rational then so would be $\sqrt{2}$).

markvs
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You’ve done the contra positive correctly, but step back and think.

$x+y + x -y = 2x$ is irrational, so $x$ must be irrational. And similarly, $y$ must be.

So if it’s true, $x$ and $y$ are irrational. If we’re to prove true, we should aim for there.

Try $x=y= \sqrt{2}$?

mdave16
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