1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Inequality proof help

  1. Oct 5, 2013 #1
    Using only the axioms of arithmetic and order, show that:

    for all x,y satisfy 0≤x, 0≤y and x≤y, then x.x ≤ y.y

    I'm really lost on where to start, my attempt so far was this

    as 0 <= x and 0 <= y, we have 0 <= xy from axiom (for all x,y,z x<=y and 0<=z, then x.z <=y.z). then we use the same axiom to get y.y >= 0, but not sure where to go from there
     
  2. jcsd
  3. Oct 5, 2013 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    There are several versions of the "axioms", so you need to tell us exactly what axioms you are given.
     
  4. Oct 5, 2013 #3
    Do you see how your axiom, when applied to the hypothesis that 0≤x, 0≤y and x≤y, gives you both x.x≤y.x and x.y≤y.y?

    Do you see where to go from here?
     
  5. Oct 6, 2013 #4
    Thanks I got it from there.

    How would I go about proving it the other way? I.e. if 0 <= x, 0 <= y, x.x <= y.y then x <= y? How would I prove that? I have gotten to (x-y).(x-y) <= 0 but not sure how to actually prove it using the axioms which are:

    A1 (x+y)+z = x+(y+z)
    A2 x + y = y + x
    A3 x + 0 = x for all x
    A3 x + (-x) = 0
    A5 x.(y.z) = (x.y).z
    A6 x.y = y.x
    A7 x.1 = x
    A8 x.x^(-1) = 1
    A9 x.(y+z) = (x.y) +(x.z)

    A10 x<=y or y<=x for all x,y
    A11 if x<=y and y <=x then x = y
    A12 x<=y and y <=z then x<=z
    A13 x<=y then x + z <= y + z
    A14 x<=y and 0<=z then x.z <=y.z
     
  6. Oct 6, 2013 #5
    I think the easiest way to prove this is by contradiction.

    If you want to try this, I think it'd be easier to start with ##0\leq y^2-x^2=(y+x)(y-x)##.
     
    Last edited: Oct 6, 2013
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Inequality proof help
  1. Inequality Proof (Replies: 9)

  2. Proof for Inequality (Replies: 4)

  3. Inequality proof (Replies: 5)

  4. Proof of an Inequality (Replies: 12)

  5. Inequality proof (Replies: 12)

Loading...