Spivak Calculus - Chapter 1 Problems

[fedlibs]

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)

Homework Statement

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

Homework Equations

12 Properties of Real Numbers

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 $x^{n}<y^{n}$ or $x^{n}>y^{n}$, 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?)

 

[kru_]

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.

 

[fedlibs]

Thank you for your quick reply.

Here is a counter example to your first proof. Let y = 2, x = -2. Now y > x. But x^2 = y^2.

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.

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.

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?)

 

[fedlibs]

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.)