Thread: Help with Proof by Contradiction

The question is: "Given two sets A and B, consider the sets A - B and B - A. Are they necessarily disjoint? Give a proof or a counterexample."

I know that they are disjoint, and I understand why. My professor said the best way to do this proof is to do a proof by contradiction.

So, I started out like this:

We proceed by contradiction. Suppose A - B and B - A are not disjoint. Then, by definition, (A-B) intersect (B-A) does not equal the empty set.

I don't know where to go from here. Can I do something like, "Put A = {1,2,3} and B = {3,4,5}" and then show how it must equal the empty set, or do A and B have to be arbitrary sets (have arbitrary elements)?

Of course they must be arbitrary.

Suppose x is A\B and x is in B\A.

So, since x is in A\B, we have x is not in B. And, since x is in B\A, we have x is in B. So x is not in B and x is in B.

Originally Posted by MoeBlee

Of course they must be arbitrary.

Suppose x is A\B and x is in B\A.

So, since x is in A\B, we have x is not in B. And, since x is in B\A, we have x is in B. So x is not in B and x is in B.

Ah, thank you. I thought it would be relatively simple, I just couldn't think of the next step.