The problem statement, all variables and given/known data
Let A be a set, prove that the following statements are equivalent:
1) A is infinite
2) For every x in A, there exists a bijective function f from A to A\{x}.
3) For every {x1,...,xn} in A, there exists a bijective function from A to A\{x1,...xn}
Relevant equations
The first and only thing that comes to my mind is that (I've read this in my textbook, but I have to prove it) if A is infinite, then it admits a bijection with a proper subset; but I'm not sure if proving that would help me to automatically say that 1) implies 2) and 3) because A\{x} and A\{x1,...,xn} are proper subsets of A, but how do I know that these are the indicated proper subsets, I mean, the statement says that if a set is infinite, it admits a bijection with A proper subset, not every proper subset.
Now, when it comes to prove that 2) or 3) imply 1) I am totally stuck. So, that's all that I have, if I think of something, I'll post it here.
Let A be a set, prove that the following statements are equivalent:
1) A is infinite
2) For every x in A, there exists a bijective function f from A to A\{x}.
3) For every {x1,...,xn} in A, there exists a bijective function from A to A\{x1,...xn}
Relevant equations
The first and only thing that comes to my mind is that (I've read this in my textbook, but I have to prove it) if A is infinite, then it admits a bijection with a proper subset; but I'm not sure if proving that would help me to automatically say that 1) implies 2) and 3) because A\{x} and A\{x1,...,xn} are proper subsets of A, but how do I know that these are the indicated proper subsets, I mean, the statement says that if a set is infinite, it admits a bijection with A proper subset, not every proper subset.
Now, when it comes to prove that 2) or 3) imply 1) I am totally stuck. So, that's all that I have, if I think of something, I'll post it here.
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