I need help on proving
(1) prove that if B A and A is countable, then B is countable
(2) prove that if B A, A is infinite, and B is finite, then A\B is infinite
I don't even know where to start... THx
I need help on proving
(1) prove that if B A and A is countable, then B is countable
(2) prove that if B A, A is infinite, and B is finite, then A\B is infinite
I don't even know where to start... THx
I have a theorem which says
Suppose A is a set. The following statements are equivalent:
1. A is countable
2. Either A = empty or there is a function f:z+ -> A that is onto.
3. There is a function f:A -> z+ that is one-to-one
I am supposed to use those however I am stuck....
If is denumerable then . If then, is finite . If let be the least positive integer such that , let be the least positive integer such that and ...
Could you continue?
Regards.
Fernando Revilla
You can use the equivalent condition #3 to show (1). Namely, suppose and there exists a one-to-one function . Consider the restriction of f on B, i.e., the function whose domain is B and that acts just like f on any . Is this restriction still one-to-one?
For (2), note that . What happens when is finite?
Whom are you asking? Fernando suggests using condition #2 to prove (1), i.e., that B is countable. Going with #2, let's assume that there is an onto function and that B is nonempty, i.e., there exists some fixed . I think it is easier to construct another function as follows:could you explain more on (1) part please???
You have to check that indeed and that g is onto.
If you are going ti use #3 to show (1), then assume that there exists an . Consider such that for and is undefined otherwise. You need to show that and that g is one-to-one.