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- Thread starter WannaBe
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You surely know that any rational number is the ratio of two integers.What is the image of the function f : Z*N -> R ; f(a,b)= a/b?

I know the answer is Q (rational numbers) but I dont know how to find it.

Note that any rational can be written as an integer divided by a positive integer.

Here let \(\displaystyle \mathbb{N}=\mathbb{Z}^+\).

- Jan 29, 2012

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As Plato said, the set of rational numbers is

Now, if y is any rational number, there exist integer, m, and positive integer, n, such that y= m/n. That is precisely f(m, n) so every rational number is in the image.

Conversely, for any pair, (m, n), m an integer, n a positive integer, f(m,n)= m/n is a rational number so the image is precisely the set of rational numbers.

(This function is NOT "one-to-one", f(2, 4)= f(1, 2), but that is not relevant to this problem.)