Hi Corey,

>>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).

This is where spelling things out really becomes useful.
Your approach may work, depending on a whole lot of context
which I haven't seen, but your definition seems to mix
set theory and logic unnecessarily. I'd be much more
comfortable with just using logic.

Firstly, a little bit of context:
When you're starting to prove theorems in some domain of
mathematics, call it A, what you want to do is to define
your terms in terms of some other, more fundamental,
domain of mathematics, call it B. And furthermore,
at the beginning, when proving theorems in domain A,
you'll be using only results from domain B in your proofs.
After you've built up a body of theorems in domain A
you can start using them to prove more theorems
in domain A. If you don't do things this way, you're in
great danger of engaging in circular reasoning, and
not really proving anything at all.

Some examples from a typical mathematics curriculum are:
Point Set Topology (A="topology" B="set theory")
Probability Theory (A="probability theory" B="measure theory")
Group Theory (A="group theory" B="set theory")
Complex Analysis (A="complex analysis" B="real analysis")

In your case (where A="set theory", B="logic") I'd expect
to see definitions of set theoretic operations like
union, intersection, symmetric difference, difference
in terms of logical operations like
inclusive or (aka or), and, exclusive or (aka xor),
negation (aka not).

So, in your case, I would be working with definitions like
x is an element of A intersect B iff
x is an element of A AND x is an element of B
and then using results from logic to establish results
in set theory. A nice example would be using the fact
that "(a xor b) xor b" is the same "a" (a result that's
heavily used in computer graphics) to establish
"(A D B) D B" is the same as "A".

Incidentally, the definition I'd use for A D B is
x is an element of A D B
iff x is an element of A XOR x is an element of B.
If you don't like "xor", restate it in terms of
"not", "or", "and"
(since a XOR b iff (a OR b) AND ( NOT (a AND b))
and work from there. If you prefer to use
a XOR b iff (a AND (NOT b)) OR (b AND (NOT a))
that's also okay :-)

> And many years ago when I took Discrete Mathematics,
> this stuff would have been easy.

Well, of course. Results from logic would have been
right at your fingertips!

> And with nearly 300 math books in my room, I cannot
> seem to find the one book I need. <chuckle>

That's a lot of books! One of the reasons that I
recommended seeking out a live mathematician is that
this kind of thing typically isn't written down.
Apprenticeship is alive and well in graduate
schools of mathematics.

> Thanks for the reference material.

You're welcome.

Hope that helps,

Walter