set theory proofs

[ripple]

Hey guys,
I know this is the algebra thread, but couldn't see anywhere related to set theory concepts so thought I'd just post here.
I am looking at set theory proofs but have no idea where to even start. I have a question that reads:

For any sets A and B. Show using set theory laws that(i) (A ∩ B) ∩ A = ∅

(ii) (A ∩ B) ∩ B = B ∩ A

also had a quick question regarding wording for a set

(iii) C = {z : z =b/a, a ≥ b, a, b ∈ N}

It appears to suggest: z is equal to a fraction (b/a) where a is equal to or greater than b and a and b are natural numbers but how do I write natural numbers if z is equal to a fraction???

Thnaks in advance for any help and apologies for wrong thread, I had no idea where else to put this!

 

[stapel]

For any sets A and B. Show using set theory laws that

(i) (A ∩ B) ∩ A = ∅

Unless there's a typo in the above statement of exercise (i), you can't prove this because it isn't true. Pick any set where A and B share some elements, so A-intersect-B isn't the empty set, and taking the intersection of A with A-intersect-B will return a non-empty set. Create such sets A and B as a counter-example.

(ii) (A ∩ B) ∩ B = B ∩ A

What are the "set theory laws" that you have? (This sort of proof is usually done via element-chasing, so we'll need to see the tools you're required to use instead of that.

By the way, the equality here is what should have been the equality in exercise (i).

also had a quick question regarding wording for a set

(iii) C = {z : z =b/a, a ≥ b, a, b ∈ N}

It appears to suggest: z is equal to a fraction (b/a) where a is equal to or greater than b and a and b are natural numbers but how do I write natural numbers if z is equal to a fraction?

Why would you need to "write natural numbers"?