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Math Help - Write a negation, contrapositive, converse

  1. #1
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    Write a negation, contrapositive, converse

    Statement: For every rational number z and every irrational number x, there exists a unique irrational number y such that z=x+y.


    Write its negation, contrapositive and converse. Simple enough right? NOT SO, according to my teacher.

    I think whats messing me up is "For every rational number z and every irrational number x" in the beginning.

    Anyone wanna take a crack at it? And can someone give me examples similar to this one?
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  2. #2
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    Re: Write a negation, contrapositive, converse

    Think of the statement as an implication, and it is easier. A is the stuff before the comma, B is after it. Then the statement is claiming "if you have case A, then case B is true".

    Negation / inverse is the negation of A implies B. Converse is B implies A. Contrapositive is both of these together. You should be able to follow from that, but let me know.
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  3. #3
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    Re: Write a negation, contrapositive, converse

    First, it is important that the original statement

    \forall z\in\mathbb{Q}\,\forall x\in\mathbb{R}\setminus\mathbb{Q}\,\exists!y\in \mathbb{R}\setminus\mathbb{Q}\;z=x+y

    uses the uniqueness quantifier. To negate the statement properly, it helps to express the uniqueness quantifier through ordinary quantifiers.

    I am not sure about contrapositive and converse, though. Strictly speaking, these concepts apply to implications. The original statement can be considered as an universally quantified implication \forall z\,z\in\mathbb{Q}\to\dots, but then one can also choose to apply contrapositive to inner implication \forall z\,z\in\mathbb{Q}\to(\forall x\,x\in\mathbb{R}\setminus\mathbb{Q}\to\dots). Also, an assumption like x\in A for some domain A is usually considered almost a part of the quantifier. For example, we usually say that the negation of \exists n\in\mathbb{N}\,n<0 is \forall n\in\mathbb{N}\,\neg n<0 and not \forall n\,n\notin\mathbb{N}\lor\neg n<0. Similarly, the usual contrapositive of \forall n\in\mathbb{N}\,n\le0\to n=0 is \forall n\in\mathbb{N}\,n\ne0\to\neg n\le0 and not \forall n\,\neg(n\le0\to n=0)\to n\notin\mathbb{N}.

    Here, when universal quantifiers are stripped from the original statement, we have an existential quantifier, and under it we have a conjunction (or biconditional), not an implication.

    Quote Originally Posted by Annatala View Post
    Think of the statement as an implication, and it is easier. A is the stuff before the comma, B is after it. Then the statement is claiming "if you have case A, then case B is true".

    Negation / inverse is the negation of A implies B.
    Hmm, do you mean that the negation of A\to B is \neg(A\to B)? This is not saying much. The negation of A\to B is A\land\neg B. Also, inverse is not the same as negation.
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    Re: Write a negation, contrapositive, converse

    Quote Originally Posted by emakarov View Post
    Hmm, do you mean that the negation of A\to B is \neg(A\to B)? This is not saying much. The negation of A\to B is A\land\neg B. Also, inverse is not the same as negation.
    Ah, my bad. It's been a looooong time since I've used those terms and I'm a little rusty. Inverse is applying negation to each side of the implication, then. That makes more sense as the contrapositive needs to be logically equivalent to the original.
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  5. #5
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    Re: Write a negation, contrapositive, converse

    It's very easy to get confused. It seems so simple though.
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