Another Proof by Induction Question

In summary: So, I am going to try and solve this without using the theorem and just the axioms. This is what I have so far:From A5 we have x + (-x) = 0. Then (-x) + x = 0 by A2. Hence x = - (-x) by the uniqueness of -(-x) in A5. From A2 we have x + (-x) = 0 and (-x) + x = 0. Hence x + 0 = 0 + x by A2. Hence x = 0 + x by A5. Hence x = x by A4. Q.E.D.Is this valid? Or am I missing something?Thank you for the help all
  • #1
mliuzzolino
58
0

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.
 
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  • #2
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
Yes, your approach is fine (if you have proved the lemma). It has nothing to do with induction though.
 
  • #4
hi mliuzzolino! :smile:
mliuzzolino said:
Is my approach a viable proof?

your proof is valid …

but it's far too complicated :redface:

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) ? :wink:
 
  • #5
1MileCrash said:
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.
 
  • #6
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!
 

1. How does induction work in proofs?

Induction is a mathematical technique used to prove statements about natural numbers. It involves showing that a statement is true for a base case, and then proving that if it is true for any given number, it must also be true for the next number.

2. What is the difference between strong and weak induction?

In strong induction, we assume that the statement is true for all numbers up to and including the given number, while in weak induction, we only assume that it is true for the previous number. Strong induction is more powerful and can be used in cases where weak induction cannot.

3. Can induction be used to prove any statement?

No, induction can only be used to prove statements about natural numbers. It cannot be used for other types of numbers, such as real numbers.

4. How do you choose a good base case for an induction proof?

A good base case should be the smallest possible number for which the statement is true. This ensures that the proof will work for all numbers larger than the base case.

5. Are there any common mistakes to avoid when using induction?

One common mistake is assuming that the statement is true for all numbers without properly proving it. It is also important to make sure that the inductive step is valid and that the base case is chosen correctly. Additionally, it is important to clearly state and explain each step of the proof to avoid any errors or confusion.

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