Understanding Set Theory: Query on +, -, *. /

[nigelwu]

Hi, I am currently reading something on Set Theory (I am not a student BTW) and I got struck. Please would somebody could give me some advices. Thanks in advance.


Is multiplication a one to one onto function G:NxN->N or G:RxR->R
I guess not. Since G(2,3)=6 and G(1,6)=6. So if not, then does this mean that there exist no inverse G such that G o InverseG = 1?
If so, does this inverseG refer to our usual sense for "division"?

Is the usual additional a one to one onto function F:NxN->N or F:RxR->R


So how do we see the addition and multiplication in the sense of Set Theory?

Does this mean that I have to stick to the symmetry, distributive, .. axioms etc.? If so why symmetry? and why distributive?...

 

[AKG]

Multiplication is not 1-1, so it has no left inverse. It has many right inverses, but since it has no left inverse, it has no inverse. If G were to have an inverse, it would be a function \mathbb{R} \to \mathbb{R}\times \mathbb{R}. Division, on the other hand, is a (partial) function \mathbb{R} \times \mathbb{R} \to \mathbb{R}, so not only is division not the inverse of multiplication, it isn't even the right type of function.

All four basic arithmetic operations are onto.

Addition is not one-to-one.

I can't make sense of the last four questions.

 

[nigelwu]

Thanks for your help. Really appreciated.

So, how would we related multiplication and division, particularly if we are doing arithematic, say x*y=z implies x=y/z for z not equal 0.

(ps since I read something saying that think of minus is a reverse process of addition)

 

[matt grime]

You are not appreciating the fact that given two numbers such that x+y=z ro xy=z then this in no way determines x or y.

 

[StatusX]

You could define a pair of functions, R->R, by f_a(b)=ab and g_a(b)=b/a for any nonzero number a, and these would be inverses.