Deriving statements using the axioms of real numbers

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    I've been asked to derive the following using the axioms of the real numbers:

    0<x<y  \Rightarrow \frac{1}{y} <\frac{1}{x}

    0<x<y , 0<z<w \Rightarrow xz<yw

    x<0 , y<0 \Rightarrow 0<xy

    I've managed to get through proving ones like  x+x=2x but the inequalities are confusing me. I'd simply like a starting point for each and any explanations as to how I can improve my thought processes, i.e. help beginning to "think like a mathematician".

    The Axioms I've been given are below

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  1. File Type: pdf LectureNotesCalculus1-6.pdf (182 Bytes, 8 views)
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    You need to post the axioms you've been given for the real numbers (there isn't one "standard set" of axioms AFAIK).
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    (Original post by DFranklin)
    You need to post the axioms you've been given for the real numbers (there isn't one "standard set" of axioms AFAIK).
    Ah, wasn't aware of this, I've updated the OP
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    (Original post by TajwarC)
    Ah, wasn't aware of this, I've updated the OP
    Unfortunately, that PDF doesn't open (for me, at any rate).
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    (Original post by DFranklin)
    Unfortunately, that PDF doesn't open (for me, at any rate).
    Apologies, I've uploaded a screenshot into the OP now
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    (Original post by TajwarC)
    I've been asked to derive the following using the axioms of the real numbers:

    0<x<y  \Rightarrow \frac{1}{y} <\frac{1}{x}

    0<x<y , 0<z<w \Rightarrow xz<yw

    x<0 , y<0 \Rightarrow 0<xy

    I've managed to get through proving ones like  x+x=2x but the inequalities are confusing me. I'd simply like a starting point for each and any explanations as to how I can improve my thought processes, i.e. help beginning to "think like a mathematician".
    Given your axioms, I think that you need to prove quite a number of intermediate results. You may already have done some of this in lectures, of course.

    For example for the final part, you could show that:

    1) x < 0 \Rightarrow -x > 0
    2) x >  0, y > 0  \Rightarrow 0 < xy
    3) (-x)(-y) = xy

    or something along those lines.
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    (Original post by atsruser)
    Given your axioms, I think that you need to prove quite a number of intermediate results. You may already have done some of this in lectures, of course.

    For example for the final part, you could show that:

    1) x < 0 \Rightarrow -x > 0
    2) x >  0, y > 0  \Rightarrow 0 < xy
    3) (-x)(-y) = xy

    or something along those lines.
    Ah I'm seeing why you've considered -x and -y. Should be enough to get me going for this one, thanks.
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    (Original post by TajwarC)
    Ah I'm seeing why you've considered -x and -y. Should be enough to get me going for this one, thanks.
    My pleasure.

    You may find it useful to prove the following for part 1):

    x > 0 \Rightarrow \frac{1}{x} > 0

    (Use trichotomy and contradiction)
 
 
 
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