Walter Vannini wrote:
>
> 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.


Well, the symmetric difference of A and B denoted (A D B) Is all x in
(A but not in B) union all x in (B but not in A).

So, if A = {1, 2, 3} and B = {2, 3, 4}

Then A D B = {1, 4}


And many years ago when I took Discrete Mathematics, this stuff would
have been easy. But, I forgot a lot of those important identities. And
with nearly 300 math books in my room, I cannot seem to find the one
book I need. <chuckle>


>
> 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.


Hmmm! I think that one is resolved by saying,

If x is an element of [A - (A - B)]

Then x is an element of A but x is not an element of (A - B)

Furthermore, x is an element of A and x is an element of (~a union B)

And this is logically equivelant to saying,

x is an element of (A intersect ~A) union x is an element of (A
intersect B)

Since A intersect ~A is the empty set, we are just left with x is an
element of the empty set union x is an element of (A intersect B)

Consequently, x is an element of (A intersect B).

So, we just start from this conclusion and prove that it is true in
reverse to demonstrate logical equivalance.

>
> 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".


I'll have to see if I can find that. Thanks for the reference
material.


Corey...