# Basic Cardinal Arithmetic

1. Mar 15, 2009

1. The problem statement, all variables and given/known data

prove that (a x b)$$^{}c$$ = (a$$^{}c$$ x b$$^{}c$$ where a,b,c are any cardinal numbers

2. Relevant equations

3. The attempt at a solution

i know that they should first be interpreted as sets A,B,C but what functions should I use.

2. Mar 16, 2009

### tiny-tim

(use the X2 tag just above the Reply box, instead of tex )
AC is the set of functions from C to A …

so pick a typical function on one side of the equation and show how to define a corresponding function on the other side

3. Mar 16, 2009

Hi and thanks but im still a bit confused so is this how i shld do it:

A^c is defined as f: C --> A and I also define B^c as g: C --> B and h can be defined as h: C --> A X B, and since assuming f and g are bijections h too is a bijection. Am I in the right direction?

4. Mar 16, 2009

### tiny-tim

No, not at all …

AC is the set of all functions from C to A, not one function

5. Mar 16, 2009

6. Mar 17, 2009

### tiny-tim

Start:

Let f:A → C be a member of AC and g:B → C be a member of BC …​

and then construct a member h:AxB → C of (AxB)C using f and g

7. Mar 17, 2009

is h going to be like this h(a,b) = (f(a),g(b) next to show that this is an injection?

8. Mar 18, 2009

### tiny-tim

Yes, that's exactly right!

('cept you missed out a bracket! )

ok, now the other way round …

starting with an h, how do you define an f and g?

9. Mar 18, 2009

I dont know how to do define an f and g starting with an h?

10. Mar 18, 2009

### tiny-tim

Hint: if h:C→ AxB is a member of (AxB)C, then define the projections hA:C→ B and hB:C→ A

11. Mar 18, 2009