Hi everybody,

I need to prove the following sets are infinite sets by finding a countable in everyone of them:

a)$\displaystyle \mathbb{Q}\backslash\mathbb{Z}$

b)$\displaystyle \mathbb{R}\backslash\mathbb{Q}$

c)$\displaystyle \{0,1\}^\mathbb{N}$

d)$\displaystyle \mathbb{N}^{\{0,1\}}$

How can I do this?

I was wondering if the following would be correct:

for example in (a): $\displaystyle \mathbb{A}\subset\mathbb{Q}\backslash\mathbb{Z}$,

where $\displaystyle \mathbb{A}= \{ x | x = \frac{1}{n}, & \forall n\in \mathbb{N} \}$.

Thanks in advance for the help.