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
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
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.