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Spivak Calculus - Chapter 1 Problems

  1. Apr 7, 2012 #1
    I have begun in my spare time to work through Spivak's Calculus. Although I have been stuck on many problems, I am most troubled by my general clumsiness and nonelegance of answers (particularly proofs). It always seems that there likely is a much simpler route available, and yet I forgo this route for a long-winded tedious argument. (Perhaps of my lack of proof experience)

    1. The problem statement, all variables and given/known data

    6 c) Prove that if [itex]x^{n}=y^{n}[/itex] and n is odd, then [itex]x=y[/itex].
    6 c) Prove that if [itex]x^{n}=y^{n}[/itex] and n is even, then [itex]x=y or x=-y[/itex].

    2. Relevant equations

    12 Properties of Real Numbers

    3. The attempt at a solution

    For 6c, I considered using the contrapositive and claiming that:
    If y≠x, then either y>x or x>y, which implies [itex]x^{n}<y^{n}[/itex] or [itex]x^{n}>y^{n}[/itex], which should complete the proof? (This was proven as the first part of the problem.)

    However, I tried to use a more direct proof from considering the factorization of the terms. This required a long winded explanation that required several rewritings for different cases, and an absolute value claim that I believe to be non-rigorous.

    Is there a simple way to do these proofs directly? (And is the contrapositive proof I provided sound?)
     
  2. jcsd
  3. Apr 7, 2012 #2
    Here is a counter example to your first proof. Let y = 2, x = -2. Now y > x. But x^2 = y^2.


    How about if x^n = y^n and n odd, then divide both sides by x^n, factor the n, and show x/y = 1 which leads to x = y. Then make a special case for when x is 0.

    Similarly for n even.
     
  4. Apr 7, 2012 #3
    Thank you for your quick reply.

    Sorry about my inclarity. The proof I provided was only for problem 6c) where n is odd case. I realize this will not work for the even case.


    Interesting, thank you for this. I have just been somewhat confused about what facts I am and am not allowed to use. Thus, I was generally hesitant with the general exponent rules. (Should I think this way?)
     
  5. May 19, 2012 #4
    So, is there a particularly clean way to show these through the factorization?

    (I am uneasy about using the exponent rules at this point of the book.)
     
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