Another Proof by Induction Question

[mliuzzolino]

Homework Statement

Prove -(-x) = x.

Homework Equations

A2: x + y = y + x [additive commutativity]
A5: x + (-x) = 0
M3: x(yz) = (xy)z [multiplicative associativity]
M4: x (1) = x
Lemma: (-1)(-1) = 1
Theorem c: (-1)x = -x

The Attempt at a Solution

-(-x) = (-1)[(-1)x] by Theorem c

= [(-1)(-1)]x by M3

= (1)x by lemma

= x by M4

Q.E.D.



This is what my approach was; however, the solution in the back of the book was something like this:

From A5 we have x + (-x) = 0. Then (-x) + x = 0 by A2. Hence x = - (-x) by the uniqueness of -(-x) in A5.

Q.E.D.


Is my approach a viable proof? I thought I was starting to understand this, but when I see the solution it completely throws me off. Additionally, the proof given by the book makes no sense to me. I have no intuition for this.

 

[1MileCrash]

All they are saying is that since (-x) + x = 0, subtract (-x) from both sides of the equation and the result is immediate.

This is not a proof by induction, by the way.

 

[Fredrik]

Yes, your approach is fine (if you have proved the lemma). It has nothing to do with induction though.

 

[tiny-tim]

hi mliuzzolino!

Is my approach a viable proof?

your proof is valid …

but it's far too complicated ​

in maths theorems, you get more marks for simpler proofs

you've used two theorems to prove it, but it could have been proved using no theorems, and two axioms

I thought I was starting to understand this, but when I see the solution it completely throws me off. Additionally, the proof given by the book makes no sense to me. I have no intuition for this.

hint: what is the definition of -(-x) ?

 

[Fredrik]

All they are saying is that since (-x) + x = 0, subtract (-x) from both sides of the equation and the result is immediate.

I think it's even more immediate than that. They're saying that by definition of "additive inverse" (see below), we can immediately conclude that the additive inverse of -x (which is denoted by -(-x)) is x.

Definition of additive inverse: For each real number y there's a real number z such that y+z=0. This z is said to be the additive inverse of y, and is denoted by -y.

 

[mliuzzolino]

Ah, my apologies. I don't know why I put induction in the title. My brain is going haywire - exam is tomorrow and this material is very hard won for me. Thanks for the clarification everyone!