# Math Help - Abstract - Prove (A-B)union(B-A)=(AunionB)-(AintersectB)

1. ## Abstract - Prove (A-B)union(B-A)=(AunionB)-(AintersectB)

Prove:
(A-B)union(B-A)=(AunionB)-(AintersectB).

We need to show (A-B)union(B-A)is contained in(AunionB)-(AintersectB)
and (AunionB)-(AintersectB)is a superset of (A-B)union(B-A).

We begin by showing the first:
Let x be an element of(A-B)union(B-A).
By definition of union, x is inA-B or xis not in B-A.
If xis in A-B, we know xis an element of A and x is not an element of B.

This is where I've begun to get stuck. Not sure where to go next.

2. I think you need to clean up your work a bit. You're trying to show that

$(A-B)\cup(B-A)=(A\cup B)-(A\cap B).$

Incidentally, both sides of this equation represent the symmetric difference of $A$ and $B$.

What you need to show is that each side is a subset of the other. The statement you wrote

We need to show $(A-B)\cup(B-A)$ is contained in $(A\cup B)-(A\cap B)$
and $(A\cup B)-(A\cap B)$ is a superset of $(A-B)\cup(B-A).$
is incorrect. The correct statement would be

We need to show $(A-B)\cup(B-A)$ is contained in $(A\cup B)-(A\cap B)$
and $(A\cup B)-(A\cap B)$ is contained in $(A-B)\cup(B-A).$
For the forward direction, you start out correctly:

Let $x$ be an element of $(A-B)\cup(B-A)$.
But then you say this:

By definition of union, $x$ is in $A-B$ or $x$ is not in $B-A$.
That is incorrect. It should be this:

By definition of union, $x$ is in $A-B$ or $x$ is in $B-A$.
Can you continue from here, or are you still stuck?

3. Originally Posted by Ackbeet
I think you need to clean up your work a bit. You're trying to show that

$(A-B)\cup(B-A)=(A\cup B)-(A\cap B).$

Incidentally, both sides of this equation represent the symmetric difference of $A$ and $B$.

What you need to show is that each side is a subset of the other. The statement you wrote

is incorrect. The correct statement would be

For the forward direction, you start out correctly:

But then you say this:

That is incorrect. It should be this:

Can you continue from here, or are you still stuck?
I always find that the easiest way to make sure you are not making mistakes with set theory stuff is to draw Venn-diagrams. They genuinely let you see what's going on.

4. ## Thanks!!!!!!

I am absoluteley delighted with the posts by kathrynmath and Ackbeet. I have touched on set theory and never came across a logical indication of how to attack the identities. All that is mentioned are Venn diagrams. The answers to the problems were never given. Maddening. "Each side is a subset of the other." tah dahh!

Thanks Swlabr for your contribution, but the Venn diagrams, though obvious, can get messy and they really are not in the spirit of a mathematical proof. But you do say they are useful as a check.