1st Year Discrete Mathematics Help

[Kevin248]

I'm about halfway through a discrete mathematics course, so this is all relatively new to me. I've also been out of high school for 7 years, so the skills are somewhat rusty as well. Anyway, onto the question:

Define a function g: Z+ x Z+ --> Z+ by the formula g(m, n) = (2^m)(3^n) for all m, n in the set Z+ x Z+.

I have to show that g is one-to-one, which I think I did, although my proof assumes that 2^x = 3^y only if x and y both equals 0, but doesn't prove this, so I don't know if that's really acceptable.

The real problem is the second half though. I need to use the above result to prove that Z+ x Z+ is countable. So I need to show that the function is onto, but I'm really lost as to where to even start.

I tried algebraically, and ended with values for m and n that I'm not sure I can prove are integers and that are defined in terms of each other, which I'm also not sure is acceptable.

This is one of those textbook exercises without an answer in the back. It just has a hint. They want me to use the fact that a subset of any countable set is countable, but I'm not really sure what to do with the hint.

[J_u_s_t_y_n_a]

Replying partially to your post; that is, for a part "So I need to show that the function is onto": It seems there is no need to prove this. If you already proved that the function is injective, then, it is ONTO some subset Y of Z+. If we use the property that a subset of any countable set is countable, we conclude that Y is countable as a subset of Z+. Then from injectivity, we can derive conclusion about Z+ x Z+ being countable.