Let X be a set and 2^X denote the powerset of X. Define a binary operation Delta on 2^X as follows:

A Delta B = (A unio...

Question

Let X be a set and 2^X denote the powerset of X. Define a binary operation Delta on 2^X as follows:

A Delta B = (A union B) \ (A intersect B)

Let H = (2^X, Delta). Which of the following statements about H is/are correct?

A. H is a group.
B. H has an inverse, but is NOT a group.
C. Every element in H has an inverse, but the inverse of A is the complement of A.
D. For every A in 2^X, the inverse of A is A.

Accepted Answer

A, D The operation Delta is the symmetric difference: A Delta B = elements in A or B but not both. This operation turns 2^X into an abelian group. The identity element is the empty set (A Delta {} = A). Each element A is its own inverse because A Delta A = {} (the empty set). So H is a group (statement A is true), and the inverse of every A is A itself (statement D is true). Statement C is wrong because the inverse is A itself, not the complement of A.