The binary operation

*

for combining sets is defined as

A * B = (A \cup B) - (A \cap B)

Prove that $G$ , consisting of the set of all subsets of a set $S$ (including the empty set and the set $S$ itself), together with the binary operation $*$ , forms a group. You may assume that the associative property is satisfied.

Consider the set of all subsets of the natural numbers and solve the equation ${1, 2, 4} * X = {3, 4}$ .