Question

The cardinality of ℤ is aleph null. Define set A, which also has cardinality aleph null, such that:
|Z - A| = aleph null.
Define set B, which also has cardinality aleph null, such that:
|Z - B| = 5.

Answer 1

I think that set a could be either Z+ or Z-, because then the result of the difference is stil aleph null. But I am not sure of the form I should answer in (what kind of work to show, or reasoning to use) and I am not sure at all how to say the difference between Z and B is 5. The only thing that comes to mind is Z = B =/= 5, but that doesn't seem right to me, it seems to easy.

Answer 2

How about defining \(B:=\mathbb{Z}-\{1,2,3,4,5\}\)

Answer 3

then \(\mathbb{Z}-B=\{1,2,3,4,5\}\) and \(|\mathbb{Z}-B|=5\)

Answer 4

as you have written \(\mathbb{Z}^+=\mathbb{N}\) is a good choice for \(A\)

Answer 5

\(\mathbb{Z}-\mathbb{Z}^+=\{\cdots,-3,-2,-1\}\) which is a countable infinite set so its cardinality is \(\aleph_0\)

Answer 6

forgot my zero

\[\mathbb{Z}-\mathbb{Z}^+=\{\cdots,-3,-2,-1,0\}\]

Answer 7

So defining B as the set of 1 to 5 gives cardinality 5? That works even though B has cardinality aleph null, but the set B would only have 5 members? Can you explain that a little for me please?

Answer 8

oooops, never mind, I see what I missed! ^_^' eheh, Thanks You very much for the help!