Is it true if the set X is countably infinite then there are bijective maps f: X-->N and f:N-->X ?

Separately, could someone start me off on this proof:

Prove that if A in uncountable, then A - {x} is uncountable.

Thanks

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- February 9th 2010, 11:54 AMChizumcountably infinite / uncountable sets
Is it true if the set X is countably infinite then there are bijective maps f: X-->N and f:N-->X ?

Separately, could someone start me off on this proof:

Prove that if A in uncountable, then A - {x} is uncountable.

Thanks - February 9th 2010, 12:08 PMPlato
- February 9th 2010, 12:15 PMChizum
- February 9th 2010, 12:37 PMMoeBlee
- February 9th 2010, 12:39 PMMoeBlee
- February 9th 2010, 01:18 PMDrexel28
Suppose that was uncountable but was countable. Then, there exists a bijection and so given by . This is clearly a bijeciton and thus a contradiction.

In fact, there is a much stronger statement that can be made.

Let be uncountable and be countable. Then is uncountable.

Proof: Since is countable there exists some which is bijective. Also, since is countable there is some which is also bijective. Clearly then, given by is a bijection, this of course contradicts that is uncountable.