5.10 Exercise: Give examples of infinitely many infinite sets no two of which have the same cardinal number.

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My text defines "countably infinite" to be a set that is countable but not infinite. Also it defines "countable" as the set is either finite or it has the same cardinal number as N, the natural numbers.

So if two sets do not have the same cardinal number, then they do not have a one-to-one correspondence, thus there exists not way to map f:A->B such that there exists at most one f(a) in B for a in A.

In previous theorems the text claims that fields N, Z, and Q are countably infinite, meaning they are not infinite sets. All of these fields certainly have an infinitely amount of terms. This doesn't make sense to me. Is the only infinite set R? I don't know if these definitions are standard, but I'm really lost obviously.

As for an example of two infinite sets that don't have the same cardinal number I actually can't think of an example because I don't know how to define an infinite set by these restrictions. It seems both sets A and B would have to have elements strictly in R, and I can't think of a way that you can't map two subsets of R and not be one-to-one.

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Basically I'm really lost and a little clarification would be a lifesaver. Thanks guys.