Question about a trivial proof

[mindCrime]

Homework Statement

I'm working on a problem in my MAT-095: Algebraic Concepts class. The overall problem is to prove the Midpoint Formula for finding the midpoint between two points on a line. But in the process of working through the proof I ran into something that I don't really remember very well from the more basic classes... I thought it was true, so I decided to try and prove it to myself. And I came up with something that "proves" what I thought I knew, but I suck at proofs so I wanted to get a little "sanity check" if I could.

Homework Equations

( x - (x + y) ) = -y

I said this was trivial. Heh.

The Attempt at a Solution

subtract x from both sides, yielding

-(x + y) = (-y -x)

distribute out a -1 from the right hand side yielding

-(x + y) = -1(y + x )

fill in the assumed 1 on the left hand side yielding

-1(x + y) = -1( y + x)

divide both sides by -1 yielding

(x + y) = (y + x), which is true by the commutative property of addition.

So, am I right, or did I miss something? Feel free to make fun of my ignorance, my math "skills" are pretty pathetic.

 

[Hurkyl]

First off, you've made a common mistake: your proof is backwards! If we assume all of your steps are correct, then you have assumed the truth of the identity

( x - (x + y) ) = -y

and used that to prove the commutative law of addition.

Fortunately, this is easy to fix: every step you used is reversible; you can apply them in opposite order to prove your identity.


Once you make that fix, it is certainly something I would normally accept as proof. I'm not sure just how pedantic you wanted to be, though.

Incidentally, you can do a proof by "calculation":

x - (x+y) = x + -(x+y) = x + (-1)(x + y) = x + ((-1)x + (-1)y)
= x + (-x + -y) = (x + -x) + -y = 0 + -y = -y

I've tried to write it overly pedantically: the only properties I used are the axioms of a ring, the definition of subtraction, and the identity -x = (-1)x.

 

[mindCrime]

Hurkly,

Thanks. I'm pretty iffy on the whole proof thing.. none of the math classes I've taken, going all the way back to high-school, through college the first time, and now this go around, really talked about proofs much. So I'm kinda trying to teach myself and work at least some of the proof related problems in my text.

I have the book "Math Proofs DeMystified" by Gibilisco, any other resources you guys might recommend for helping a math n00b bridge the gap into understanding proofs and the more formal stuff? There's also some stuff in my old discrete math book on proofs, but we didn't cover it in the class, so I'll probably go back and look at that too.

Thanks again...

 

[Hurkyl]

I can't help with that. But if you go ask it in our general math forum, I'm sure others will do so.