Problem:

Let S be a set. For any two subsets of S we define

A + B = (A-B) U (B-A)

A*B = A 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 A + (B+C)

either x A or x B or x C but x belongs to only one of these sets otherwise x couldn't be in A + (B+C)

if x A then x (A+B)+C

if x B then x (B + A)+C = (A+B)+C

if x C then x doesn't belong to (A+B) and x (A+B)+C

so A + (B+C) is a subset of (A+B)+C

The same agument works for using x (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 B

if x A+B then x A+C and x doesn't belong in A so x 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 A so x C so B is a subset of C. The same agument works starting with x 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