Re: Existence of function

In article <1125528895.038360.288810@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
agapito6314@xxxxxxx wrote:

> Let J be the set of non-negative integers. Set S is the countably
> infinite union of countably infinite sets. How does one prove
> existence of function g: JxJ ----> S? Thanks.

As the others pointed out, what you probably meant to ask is, how does
one prove the existence of an *injective* function g:JxJ -> S. If all
you want is any old function, a constant function will do.

If S is a countable union of countable sets, S = A1 union A2 union A3
.... where each Ai consists of a countable collection of elements Ai_1,
Ai_2, etc. In other words any element of S is of the form An_m for some
nonnegative integers n and m.

For notational convenience we may as well dispense with the letter A,
and just consider S to be the set of pairs (n,m) where n and m are
nonnegative integers.

Now for (n,m) in S, define g(n,m) = 2^n * 3^m. For example g(5,4) = 2^5
* 3^4 = 32*81.

That's an injective function from the set of pairs (n,m) into the
positive integers. Injective means that different pairs go to different
integers, which (we suspect) is what you really meant.
.

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