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Math Help - Quantifier Proofs

  1. #1
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    Quantifier Proofs

    Are these quantifier proofs correct? You just have to take counterexamples to disprove, and examples to prove right?
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  2. #2
    MHF Contributor red_dog's Avatar
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    (i) Let x\in\mathbf{R}. Take y=-x+1\Rightarrow x+y=1>0, so the statement is true.
    (ii) Let x\in\mathbf{R}. Take y=x-1\Rightarrow x-y=1>0, so the statement is true.
    (iii) For y=x-1\Rightarrow x+y=-1<0, so the statement is false.
    (iv) For x=0\Rightarrow 0\cdot y=0,\forall y\in\mathbf{R}, so the statement is false.
    (v) If x=0\Rightarrow 0\cdot y=0,\forall y\in\mathbf{R}.
    If x\neq 0, take y=-x\Rightarrow xy=-x^2<0, so the statement is false.
    (vi) Let x\in\mathbf{R}. Take y=x\Rightarrow xy=x^2\geq 0, so the statement is true.
    (vii) Take x=0\Rightarrow 0\cdot y=0\geq 0,\forall y\in\mathbf{R}, so the statement is true.
    (viii) Take y=-x and the statement is true.
    (ix) (x+y>0) \wedge (x+y=0) is false, so the statement is false.
    (x) (\forall x\in\mathbf{R}, \exists y\in\mathbf{R},x+y>0) is true (see (i)).
    (\forall x\in\mathbf{R},\exists y\in\mathbf{R},x+y=0) is true (take y=-x), so the statement is true.
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  3. #3
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    so the way i do it (choose numbers) is wrong?
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  4. #4
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    Quote Originally Posted by shilz222 View Post
    so the way i do it (choose numbers) is wrong?
    Yes. Look at how the Dog did it it. He pick any number and should you can find a corresponding number for that one. You picked a specific number and picked a corresponing number.
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  5. #5
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    For (3) do you mean  y = -x-1 ?
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