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Groups of Sets

Age 16 to 18 Challenge Level:

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 $\{1,2,4\}*X = \{3,4\}$.