Define a one-to-one mapping from the set of natural numbers onto the set of posi

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Mathematicians have very precise definitions for terms like "infinite" and "same size". The single unambiguous correct answer to this question is that using the standard mathematical definitions, the rationals have the "same size" as the integers. First, here are the definitions: Define "0" = emptyset, "1" = {0}, "2" = {0,1}, "3" = {0,1,2}, etc. So, the number "n" is really a set with "n" elements in it. A set A is called "finite" iff there is some n and a function f:A->n which is bijective. A set A is called "infinite" iff it is not finite. (Note that this notion says nothing about "counting never stops" or anything like that.) Two sets A and B are said to have the "same size" if there is a some function f:A-> B which is a bijection. Note that we do NOT require that ALL functions be bijections, just that there is SOME bijection. Once one accepts these definitions, one can prove that the rationals and integers have the same size. One just needs to find a particular bijection between the two sets. If you don't like the one you mentioned in your post, may I suggest that Calkin-Wilf enumeration of the rationals? (Simply google search Calkin Wilf counting rationals. The first .pdf has what I'm talking about). Of course, these give bijections between the naturals (with out 0) and the rationals, but once you have a bijection like this, it's easy to construct a bijection from the integers to the rationals by composing with a bijection from the naturals to the integers. n mathematics a set is called infinite if it can be put into a 1-1 correspondence with a proper subset of it, and finite it is not infinite. (I know it seems crazy to have the concept of infinite as primitive and finite as a derivate, but it's simpler to do this, since otherwise you must assume that the integers exist before saying that a set is finite) As for your remarks: - with your method (if you don't forget to throw out fractions like 4/6 which is equal to 2/3) you actually counted the rationals, since for each number you have a function which associates it to a natural number. It's true that you cannot count ALL rationals, or all integers; but you cannot either draw a whole straight line, can you? - with infinite sets you may build infinite mappings, but you just need a single 1-1 mapping to show that two sets are equal.

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