Problem:
Show that has the same cardinality as .
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From my understanding (and using the book's definition), the cardinality is just the number of elements in the set. So I think I need to prove that there exists a one-to-one function mapping the infinitely many numbers between 0 and 1 to all real numbers .
I looked at cos(x) and sin(x) to maybe just try to say something like, function f(x) = cos(x) maps all real numbers into a number between 0 and 1, but this is not true. And even if I rigged that to work, it's not a one-to-one function since cos(x), sin(x) etc... can equal the same value with different inputs. Am I on the right track, though?
I don't want the answer, just a little guidance.
Thanks.
Thanks for the response! I was working with tan for a bit, but was discouraged about the facts of when it was undefined... I was thinking about it the wrong way.
It seems to me that it would be one-to-one since for each value of x between 0 and 1, it would give a distinct output in . I think it is also onto since, looking at the graph of this function, it spans the entire codomain off into infinity in both vertical directions.