Thread: venn diagrams + proving set examples

Draw venn diagrams to ilustrate the following, then prove them.
(i) B\A'=B∩A
(ii) (A∩B)'=A'∪B'
(iii) A∩(B∪C)=(A∩B)∪(A∩C)

Any help will be much appreciated.

Hey BlackAdders.

Do you know what the diagrams look like when drawn (not mathematically proven)? If you draw two circles that overlap within your universal set, the parts that are common between the circles corresponding to A and B is A∩B and the part that is common to both circles is everything in both A and B A∪B. You should start off by drawing a diagram of these (even in microsoft paint or some other similar program) and attach them here because this part doesn't need any mathematics at all.

Thanks for the answer. I'm pretty sure I know how to do the drawing bit. The problem is I'm not sure about mathematical proofs.
1. x ∈ B and x ∉ A'; x ∈ B and x ∈ A; so A ⊆ B
2. x ∉ A∩B; x ∈ A and x ∈ B
3. x ∈ A & B, x ∈ A & C

The first one has the definition of A\B = A and B^c, and we know that if complement something twice we get back the original. So (B^c)^c = B which means A\B^c = A and (B^c)^c = A and B.

For the second one, we know that anything plus its complement (like the above) will be the universal set or Omega. So this means that if we have a set X and its complement X' then U\X = X'.

I'm not really sure how much detail you need to go into proving these, so I might get your feedback on that. If you have to go really deep instead of being able to assume a lot of the identities, then what I said above may not be sufficient.

I believe I should use the idea that right side is a subset of the left one. And converse.

corrected diagrams.