Few beginner set theory questions I have two problems in beginner set theory, and one I am having lot of trouble with, one I am not sure and would want to know if is it correct.

The first one I am having trouble with is

(1)

Suppose $B \subset A_{1}$ and $B \subset A_{2}$

if $$|A_{1}\setminus B|=|A_{2} \setminus B|$$

Then what can be said about the statement $|A_{1}|=|A_{2}|$ 

Either it is always true, or true if $A_{1}$ and $A_{2}$ are finite but not in general, or it could be false regardless of if they are finite or not.

I keep trying to come up with counter examples but I don't seem to get anywhere, I know the definition of cardinality we are using implies there is a bijection between the sets. To me, it seems like this would be true when finite, I am having trouble finding out fersure. I think when infinite, we could probably come up with an example but I am again not sure. I also don't know how to prove such statements. If I did have to pick right now I would probably choose True if finite but not always.

Update;   But how I can say there a bijection between the set without b implies a bijection between the A?
(1) 


   No, that claim is false. Take $A_1 = \mathbb{N}, A_2 = \mathbb{R}, B = \mathbb{R}\setminus\mathbb{Q}$ (I'll assume you know $|\mathbb{N}| = |\mathbb{Q}| = \aleph_0 &lt; \aleph_1 = |\mathbb{R}|$


Oops, sorry, I missed the part where $B \subset A_1, A_2$

Then it's true, because $|X| = |Y| \iff \exists f:X \rightarrow Y$ where $f$ is bijective, and it's easy to extend the bijection on $B$ to get $f':A_1 \rightarrow A_2$ bijective.

(2) To prove equality between sets you should always proceed by showing  double inclusion, i.e. $$X = Y \iff X \subset Y \land Y \subset X$$
(1) 


   No, that claim is false. Take $A_1 = \mathbb{N}, A_2 = \mathbb{R}, B = \mathbb{R}\setminus\mathbb{Q}$ (I'll assume you know $|\mathbb{N}| = |\mathbb{Q}| = \aleph_0 &lt; \aleph_1 = |\mathbb{R}|$


Oops, sorry, I missed the part where $B \subset A_1, A_2$

Then it's true, because $|X| = |Y| \iff \exists f:X \rightarrow Y$ where $f$ is bijective, and it's easy to extend the bijection on $B$ to get $f':A_1 \rightarrow A_2$ bijective.

(2) To prove equality between sets you should always proceed by showing  double inclusion, i.e. $$X = Y \iff X \subset Y \land Y \subset X$$