- #1
mliuzzolino
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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.