Problem:

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

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

A*B = A $\displaystyle \cap$ 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 $\displaystyle \epsilon$ A + (B+C)

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

if x $\displaystyle \epsilon$ A then x $\displaystyle \epsilon$ (A+B)+C

if x $\displaystyle \epsilon$ B then x $\displaystyle \epsilon$ (B + A)+C = (A+B)+C

if x $\displaystyle \epsilon$ C then x doesn't belong to (A+B) and x $\displaystyle \epsilon$ (A+B)+C

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

The same agument works for using x $\displaystyle \epsilon$ (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 $\displaystyle \epsilon$ B

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