Giving examples of denumerable sets

Question

My question reads:

Give an example of denumerable sets A and B, neither of which is a subset of the other, such that

(a) A ∩ B is denumerable

(b) A-B is denumerable.

I am not sure if I would have to pick subsets of like the integers.

Answer 1

The most straightforward way is to use subsets of the integers (or rationals). For (a), let $A=\mathbb{N}\cup \{-1\}$ and let $B=\mathbb{N}\cup \{-2\}$. Then their intersection is denumerable. Construction like this, where their intersection is a set well known to be denumerable, is the easiest way to go. For (b), try choosing a set $A$ that mostly contains $B$. In other words, add an extra element to $B$ so that $B$ is not a subset of $A$. Make sure that the compliment of $B$ in $A$ is also denumberable. The even/odd numbers in $\mathbb{N}$ should work well for this.

Answer 2

How about:

$A$ : all multiples of 2, i.e. $A =\{0,2,4,6,...\}$

$B$ : all multiples of 3, i.e. $B =\{0,3,6,9,...\}$

Then $A \cap B$ is all multiples of 6, i.e. $A \cap B = \{0,6,12,...\}$

And $A -B$ is all multiples of 2 that are not multiples of 6, i.e. $A-B = \{2,4,8,10,14,..\}$

And clearly $A$ and $B$ are not subsets of each other.