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Math Help - Proof by contrapositive

  1. #1
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    Proof by contrapositive

    I am having a terrible time understanding this one:

    Prove that for all x E r, and y E R, if x is rational and y is irrational, then x + y is irrational. use proof by contradiction known as contrapositive.

    OK I understand that contrapositive (~p --> ~q) so

    Proof: if x is not rational then y is not irrational


    Now I have no idea what to do next?

    Please help, I have a test coming up in two days and I need to grasp this stuff.
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  2. #2
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    Quote Originally Posted by robasc View Post
    I am having a terrible time understanding this one:

    Prove that for all x E r, and y E R, if x is rational and y is irrational, then x + y is irrational. use proof by contradiction known as contrapositive.

    OK I understand that contrapositive (~p --> ~q) so

    Proof: if x is not rational then y is not irrational


    Now I have no idea what to do next?

    Please help, I have a test coming up in two days and I need to grasp this stuff.
    So assume the negation of the conclusion.

    Original conclusion: x + y is irrational
    Negation of it: x + y is rational

    So under this assumption you can write x + y = r, where r is in Q (rationals). Now what happens when you solve for y?
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  3. #3
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    I have no idea, I am assuming that the answer will be false because a rational number and a irrational number will conclude to be an irrational number but how do I prove that?
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  4. #4
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    Quote Originally Posted by robasc View Post
    I have no idea, I am assuming that the answer will be false because a rational number and a irrational number will conclude to be an irrational number but how do I prove that?
    What does "answer will be false" mean?

    x + y = r with r in \mathbb{Q}, by assumption
    \implies y = r - x

    recall that \mathbb{Q} is closed under + and -. Thus y is rational. Contradiction.
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