1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: If x^n=y^n and n is odd, then x=y.

  1. Jun 13, 2013 #1
    This is all seems fairly obvious to me and proving it is a bit awkward. In the previous exercise we proved that if x<y and n is odd, then x^n<y^n.

    Spivak argues that, from the previous exercise, x<y would imply that x^n<y^n and y<x would imply that y^n<x^n. That's his whole proof. Is he simply saying that, by analogy, this relationship must hold?

    I mean qualitatively it makes sense, because anything raised to an odd exponent retains it's original sign. Thus for x^n=y^n to hold, either x=y or -x=-y (which is obviously the same as x=y).
  2. jcsd
  3. Jun 13, 2013 #2
    if x is not equal to y then either x<y or y<x which both imply that x^n is not equal to y^n
  4. Jun 13, 2013 #3


    User Avatar
    Science Advisor

    No. What he is doing is using "trichotomy", that, for any numbers x, y, one and only one must hold:
    x= y
    x< y
    y< x.

    If x is not equal to y then either x< y or y< x. If x< y then [itex]x^2< y^2[/itex], contradicting "[itex]x^2= y^2[/itex] so that is not the case. If y< x then [itex]y^2< x^2[/itex], contradicting "[itex]x^2= y^2[/itex]. I presume that Spivak feels the "y< x" case is so similar to the "x< y" case that the reader could see that for himself.
  5. Jun 13, 2013 #4
    That makes perfect sense. I guess I'm not quite up to the level of the average reader then. My apologies.

    Thanks guys.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted