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Math Help - Relations, reflexive and transitive

  1. #1
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    Relations, reflexive and transitive

    The relation R on real numbers given by

    xRy <=> x-y E(belongs) to Q

    can you help me with Symmetric and Transitive?

    Someone told me this is how to show its symmectric

    x-y = a/b

    -(x-y) = y-x = -a/b

    cant figure out transitive?
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  2. #2
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    Here is something that you could do almost without thinking.

    The definition of a transitive relation applied to R: for any three numbers x, y, z, if x-y\in\mathbb{Q} and y-z\in\mathbb{Q}, then x-z\in\mathbb{Q}. One has to prove this. Let's fix some numbers x, y and z and assume that x-y\in\mathbb{Q} and y-z\in\mathbb{Q}. This means there are rational numbers a / b and c / d such that x - y = a / b and y - z = c / d. One has to prove that x - z is a rational number.

    Here you actually turn on the brain: x - z = (x - y) + (y - z). Is this number rational?
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  3. #3
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    I suppose, since the sum of to rational numbers will be a rational number?
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  4. #4
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    Yes.
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  5. #5
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    I was wondering if you could tell me how to see if the relation R on real numbers given by

    xRy <=> x*y > 0

    is transitive?


    I'd so no, since I can pick out numbers that will not satisfy
    x*y > 0 and y*z > 0 => x*z > 0.
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  6. #6
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    x\cdot y>0 if and only if they both positive or both negative. i.e have same sign not zero.
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  7. #7
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    So the relation is transitive? if they share same sign and one of them or both isnt zero?
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  8. #8
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    Yes it is transitive.
    But neither can be zero. Both must be positive or both negative.
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