# Thread: Addition and multiplication with Sets

1. ## Addition and multiplication with Sets

Problem:
Let S be a set. For any two subsets of S we define
A + B = (A-B) U (B-A)
A*B = A $\cap$ B
Also I know that A + B = B +A
A + {} = A
A * A = A
A + A = {}
Prove that:
a) A + (B + C) = (A+B)+C
b) If A + B = A + C then B = C
c) A * (B + C) = A*B + A*C

a) let x $\epsilon$ A + (B+C)
either x $\epsilon$ A or x $\epsilon$ B or x $\epsilon$ C but x belongs to only one of these sets otherwise x couldn't be in A + (B+C)
if x $\epsilon$ A then x $\epsilon$ (A+B)+C
if x $\epsilon$ B then x $\epsilon$ (B + A)+C = (A+B)+C
if x $\epsilon$ C then x doesn't belong to (A+B) and x $\epsilon$ (A+B)+C
so A + (B+C) is a subset of (A+B)+C
The same agument works for using x $\epsilon$ (A+B)+C and showing that (A+B)+C is a subset of A+(B+C) thus A+(B+C) = (A+B)+C

b)let x $\epsilon$ B
if x $\epsilon$ A+B then x $\epsilon$ A+C and x doesn't belong in A so x $\epsilon$ C thus B is a subset of C.
if x doesn't belong to A+B then x doesn't belong to A+C and $\epsilon$ A so x $\epsilon$C so B is a subset of C. The same agument works starting with x $\epsilon$ C so since B is a subset of C and C is a subset of B then B=C.

c) I'm not really sure where to start on this one.

Are the two proof's reasonable?
Thanks

2. The way ‘+’ is defined depends upon on the idea of “symmetric difference” of two sets.
It is well know that using that definition an algebra of set can be defined.
You need to look up the concept of “symmetric difference”.

3. The problem defines + as A + B = (A-B) U (B-A) which is consistent with one of the definition's of "symmetric difference" I found. For the problems I just thought of it as exlusive or.