So, . When talking about set operations, this is called the symmetric difference and is commonly denoted by .
Do you have difficulties with this?
Closure - show well-defined
The associativity of intersection is pretty obvious. For addition, it's a little messy. Hint: prove that iff x belongs to exactly one of a, b, c or to all three, and similarly for .
Assoc - (a+b)+c = a+(b+c) and a(bc) = (ab)c
Follows directly from the definition and commutativity of union and intersection.
Comm - a+b = b+a and ab = ba
Draw a Venn diagram of a, b, c and then prove that iff .
Dist - a(b+c) = ab+ac and (a+b)c = ac+bc