Hi Corey,

> I was not sure how to prove that ...

Wow, this is hard to answer, especially via email!

It seems to me that you really want to clarify just
what you're given, and what you have to establish.
Knowing what you're given includes knowing the
definitions of the terms you're working with.
The definition you're using for "symmetric difference"
is of course crucial.

Probably your best bet is to look at a simpler but
similar problem. One example that comes to mind is:
show that A-(A-B) is the same as A intersection B.
Whatever technique you use for proving this will
most probably apply to your problem.

If you're at a university and have access to mathematicians
(in person), I would strongly suggest asking one of them
to show you how to give a rigorous proof of
A-(A-B) = A intersection B. Incidentally, the way I'd
prove it is by showing that if x is an element of the
set on one side, then it's an element of the set on the
other side, and vice versa. For now, I'd recommend
being really pedantic and being explicit about every
step and assumption.

You should keep in mind that there are usually many ways
to prove a result, and you shouldn't stop looking once
you've found a proof.

Also, in case you haven't come across it, I'd suggest
becoming familiar with Polya's "How to Solve It".

By the way, your distrust of accepting a proof by diagram
has a long tradition. The ancient greeks were known to
deliberately draw misleading geometric diagrams so that
they would not be fooled into assuming something they
had not logically established. For example, if they were
proving something about an isosceles triangle, they might
draw a triangle with all three sides being very different
in length. Apparently they weren't careful enough, and
Hilbert found some holes in some of Euclid's proofs
that were most probably due to relying on diagrams. Things
like assuming that a point was inside a polygon simply
because it looked like it should be in the polygon.

Good luck,

Walter

--- Corey Bray wrote:
> Hello Group,
>
>
> Let D represent capital Delta and denote the symmetric difference of
> two sets A and B. Prove that,
>
> (A D B) is logically equivelant to (A union B) - (A intersect B)
>
> I understand to prove that two sets are logically equivelant means to
> show that P is a subset of Q and also Q is a subset of P. But, I was
> not sure how to prove that the above was logically equivelant. I can
> see that it is true using pictures, but I want to improve my
> understanding of how to do such proofs abstractly rather than relying on
> pictures.
>
>
> Corey...
> isomorphics@...