# Thread: Let A, B, and C be subsets of some Universal set U...

1. ## Let A, B, and C be subsets of some Universal set U...

A) Draw two general Venn diagrams for the sets A,B,and C. On one, shade the region that represents A-(B-C), and on the other, shade the region that represents (A-B)-C. Based on the Venn diagrams make a conjecture about the relationship between the sets A-(B-C) and (A-B)-C.

B) Prove the conjecture you made from Part A.

I have attempted the venn diagrams, but I am not sure if they are right, or what the relationship between the two sets are. So I cannot proceed to the proof. Also, could someone shed the light on how to do the proof once the conjecture is made?

The relationships I got are as follows:

(A-(B-C)) intersect ((A-B)-C) = ((A-B)-C)
and
(A-(B-C)) union ((A-B)-C) = (A-(B-C))

Does this sound like the correct relationship?

2. ## Re: Let A, B, and C be subsets of some Universal set U...

Did you do the Venn diagrams correctly?

3. ## Re: Let A, B, and C be subsets of some Universal set U...

Yes I did the venn diagrams correctly and they correspond to my hypothesiszed conjecture, but how do I prove it?

4. ## Re: Let A, B, and C be subsets of some Universal set U...

you could combine both conjectures into one:

(A-B)-C ⊆ A-(B-C)

(since if X ⊆ Y, XUY = Y, X∩Y = X).

then proof is a somewhat easier matter: assume x is in (A-B)-C, and show x must always be in A-(B-C).

(in words: x is in (A-B)-C means: x is in A, but not B, nor in C.

youre eventual destination is: x is in A, but not in the part of B which doesn't intersect C).

my advice is: show that (A-B)-C ⊆ A-B ⊆ A-(B-C), because the two "intermediate proofs"

(A-B)-C ⊆ A-B and A-B ⊆ A-(B-C) are easier to do (because A-B is a subset of A, and B-C is a subset of B,

use the idea: "if you take out less, you have more").