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I was given the proposition, ∃x,y ∈ R, x is irrational and y is irrational and x+y is rational To prove this, I first converted it into an if statement since it would make it easier to prove. I got, If -(∃x,y ∈ R, if X and y are irrational then X + y is irrational). From what I know, if I prove that the statement inside the brackets is false, by negating it, it would prove that the original proposition is true. However, I can't think of a way to prove this. Can someone help me out?

Shoaib Ahmed
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    For any irrational $x$, note that $x+(-x)=0$ is rational. –  Mar 02 '17 at 20:00
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    Suppose $x = \sqrt 2, y = 1-\sqrt 2$ – Doug M Mar 02 '17 at 20:00
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    In this situation, since you are proving an existence theorem, it is enough to show find an example, which will prove that such an object exist. It is important to note that an example is not a proof in general, but here it would be enough. – Maxime Scott Mar 02 '17 at 20:03
  • If -(∃x,y ∈ R, if X and y are irrational then X + y is irrational). From what I know, if I prove that the statement inside the brackets is false, by negating it, it would prove that the original proposition is true. Sorry, but this makes no sense. – dxiv Mar 02 '17 at 20:31

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