Hi, I'm a high school student currently working through Spivak for the first time and this is my first proof based introduction to mathematics so I just wanted to make sure that I was doing the following correctly:
Prove that if [itex] x^2 = y^2[/itex] then [itex]x=y[/itex] or [itex]x=-y[/itex]
The Attempt at a Solution
[itex] (-y)^2 = (-y)*(-y) = (-1*y)*(-1*y) = (-1)*(-1)*y*y = 1*y*y = y*y = y^2 [/itex]
[itex] (-y)^2 = y^2 [/itex]
[itex] x^2 = y^2 [/itex] when [itex] x= -y [/itex]
Is this proof logical assuming the 12 basic properties of multiplication and addition that spivak lists in chapter 1?
For the fist case ([itex] x= y [/itex]) I'm kind of confused. I could do this:
if [itex] y*y = y^2 [/itex]
then [itex] y^2 = y^2 [/itex] and [itex]x^2 = y^2[/itex] if [itex] x=y[/itex]
but that seems far too easy, am I missing something or is that part just that simple?
Thanks for any help :)