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Let X be a set and 2^X denote the powerset of X. Define a binary operation Δ on 2^X as follows: AΔB = (A - B) ∪ (B - A). Let H = (2^X, Δ). Which of the following statements about H is/are correct?

1. H is a group.
2. Every element in H has an inverse, but H is NOT a group.
3. For every A ∈ 2^X, the inverse of A is the complement of A.
4. For every A ∈ 2^X, the inverse of A is A.

Correct answer: For every A ∈ 2^X, the inverse of A is A.

Solution

The operation Δ defined as AΔB = (A - B) ∪ (B - A) corresponds to the symmetric difference, which is associative and has an identity element (the empty set). The inverse of any set A under this operation is A itself, since AΔA results in the empty set, confirming that each element is its own inverse.

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