Does anyone know how to write a FORMAL PROOF for (For all real numbers x)(-(-x)=x)
using the definition (For all real numbers x)(-x = 0-x)
So i need to obviously use the axioms of arithmitec etc... any1 can help me do it? it should take about 20 steps not much less but shouldntbe greater than 20
http://www.math.rutgers.edu/~sussmann/papers/hmw10.pdf is where the axiom rules can be foudn ont he last part of the page
The actual way is well-defined as the additive inverse of in any ring.
is our identity element.
And the fact that,*)
Is a theorem (called "rule of signs") hold true in any ring.
Like the ring of reals.
These are not definitions.
*)I can prove it, if you thus wish.
does anyone kno how to write the formnal proof of it using the axioms of arithmetic, it cant be 5 steps, it has to be around 20 i was starting out like
s1. Defintion given
s2. Let a be a real number that is arbitrary
s3. then -a = 0-a
s18. -a = (-a) because for all x = x equality axiom
s19. -(-a) = a
s20. then for all x belonging to real numbers such that (-(-x
it would be sumthin lioke that i figure, its all about detail and showing eacxh step, any1 can finish the puzzle?
im just using the axioms of arithmetic, i understand what you mnean with invrerse, but it ewould e wrong to provee it like that because it isnt done ussing the axioms of arithmetic, ive attached the pdf file if you need to see the axioms of arithmetic, ONLY that can be used, otherewise this would have been an easy problem
Frankly, I see nothing in the set of axioms you posted that allows this to be proved. As the Captain has already told you, this is usually done using the eight ring axioms. This is one in a sequence of theorems leading to the proof that (-a)(-b)=ab. Therefore, it is more than the inverse of an inverse is a group. It is about both multiplication and addition.
Is not an axiom.
As Plato said it satisfies the ring axioms...
1)The set is an abelian group under addition (5 axioms: closure, associate, identity, inverse, commutativity).
2)The set is closed under multication (1 axiom) and left and right distributive laws hold:
A total of 8 axioms.
That list you gave is faulty.
Too much stuff.
My own question. Why are these called "axioms"? They certainly can be proved, all of them. Is it because they are independant?
assumed symmetry of "=" again.
You should be uncomfortable with something like "x-(-x)=0" because we
have in mind this system representing arithmetic with reals (even though
they appear undefined here), so we do not want for every real x: x+x=0 which
contradicts a property of our model that we want this system of axioms to
represent at some level.
I must say that the order axioms look as though they may permit the
proof of the symmetry of "=", but it does not look nice (it looks as
though the burden is shifted to the commutativity of the logical conectives
"and" and "or".) Also it looks to me as though "=" has the same status here
as the other "logical" symbols which seem to be a bit short on defined properties,
which presumably are already defined.
i tihnk i figured out how it's suppose to be done: IM going to restate the professors question so i can show you whyi cant use anything else
WRITE A FORMAL PROOF OF THE STATEMENT (FOR ALL x BELONGING TO REAL NUMBERS)(-(-x=x), using the definition (FOR ALL x BELONGING TO REAL Numbers)(-x=0-x), the axioms of arithmetic, the equality axion (For all x)(x=x), the 17 rules of logic such as (substitution, special case rule, iff-use, witnessrule, etc..), and NOTHING ELSE.
This is how i approached it, tell me what you guys think, i know its a stupid way the professor wants it done but thats his rules lol well here i go:
S1. (FOR ALL X BELONGING TO REAL Numbers)(-x=0-x) [Definition given]
2. (For all x)(x=x)
3. Let a be an arbitrary real number
4. a=a using step 2
5. 0 belongs to the set of integers NZ3 Axiom
6. then a - a = 0 via Axion SUB2
7. a+(0-a)=0 Substitution of definition in step 1
8. a = -(0-a)
10. -(-a) - a = 0
11. from step 6 we substitute 0 ith a-a which gives us -(-a)-a=a-a
14. (For all x)(For y)(x+y=y+x)
15. So from step 14 0+a=a+0
16. (For all x belonging to real numbers)x+0=x
17. Then -(-a)=a+0
18. from step 16 -(-a)=a
19. Hence (for all x belonging to real numbers)(-(-x)=x