# Another Proof by Induction Question

1. Apr 14, 2013

### mliuzzolino

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

Prove -(-x) = x.

2. Relevant 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

3. 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.

2. Apr 14, 2013

### 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.

3. Apr 14, 2013

### Fredrik

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

4. Apr 14, 2013

### tiny-tim

hi mliuzzolino!

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

5. Apr 14, 2013

### Fredrik

Staff Emeritus
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.

6. Apr 14, 2013

### 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!