how to prove X^2 > 0 or X^2 = 0

[farfar19]

i don't know how to prove X^2>0 or X^2=0 based ON the axiom list! it is a differential calculus course that contains basic proofing

 

[pka]

Re: how to prove X^2>0 or X^2=0

i don't know how to prove X^2>0 or X^2=0 based ON the axiom list! it is a differential calculus course that contains basic proofing

Has it occured to you that we have no idea what your axiom list is?

 

[stapel]

i don't know how to prove X^2>0 or X^2=0 based ON the axiom list! it is a differential calculus course....

I will guess that "it is" means "This exercise was assigned in". But I'm afraid I don't understand the capitalization of the word "on"...? You don't bother capitalizing the pronoun "I" or the first words of sentences, which implies that this particular capitalization must define or stand for something of some significance. But I'm afraid I can't think of anything that it could be...?

Please reply with clarification, including (as requested earlier) the complete list of axioms from which you are supposed to be working. The exact text of and instructions for this exercise (meaning "word for word, in standard English") would also be very helpful.

Thank you.

Eliz.

 

[farfar19]

The axioms for the real numbers

We assume the existence of a set R , called the real numbers, that satisfies the following list of axioms. There are functions

+:RXR--->R, .:RXR--->R

I have to prove that for every set of X that is contained in R , X square is bigger and equal to zero.

Some of the axioms are:

(a+b)+c=a+(b+c)
ab=ba

if a<b and b<c then a<c;
if a<b then a+c<b+c;
if a<b and 0<c then ac<bc;

Please let me know if you need more information.

 

[stapel]

Have you any axioms related specifically to zero or to one? For instance, have you anything that says something like "For a real number m and the additive identity 0, it is always true that 0×m = 0"?

Also, how are "+" ("plus") and "." ("point") defined? (You listed them as being functions, but have they any defined behaviors?)

Thank you!

Eliz.

 

[farfar19]

0.X=X has been proved before and I have all of the steps. and I proved this by using the axioms lists.

I only have bellow axioms related to Zero

there exists unique element 0 is contained in R such that for every a,a+0=a;

and

for every a there exists unique element -a such the a+(-a)=0;

and

if a is not equal to zero , there exists a unique element a to power of -1 such that aa to power of -1 equal to a;

Thanks

 

[stapel]

Have you any axiom related to distributivity? Have you any axioms related to "1"?

How are the functions "+" ("plus") and "." ("point") defined?

Thank you!

Eliz.

 

[pka]

Why not simply write out the complete set of axioms?
Doing anything half-way does not get you anywhere.

 

[Opalg]

i don't know how to prove X^2>0 or X^2=0 based ON the axiom list!

One of the axioms should say something to the effect that for any real number x, either x<0 or x=0 or x>0. Look at each of these cases separately, and use axioms such as the ones you quoted:

if a<b and b<c then a<c;
if a<b then a+c<b+c;
if a<b and 0<c then ac<bc;

 

[daon]

One of the axioms should say something to the effect that for any real number x, either x<0 or x=0 or x>0.

That may or may not be an axiom. I remember proving the order trichotomy as a result of more primitive axioms. It was stated with the additional property that at most one of them holds.