Let X be a set and 2X denote the powerset of X. Define a binary operation Δ on…
2023
Let X be a set and 2X denote the powerset of X.
Define a binary operation Δ on 2X as follows:
\(AΔB = (A − B) ∪ (B − A) .\)
Let H = (2X, Δ). Which of the following statements about H is/are correct?
- A.
H is a group.
- B.
Every element in H has an inverse, but H is NOT a group.
- C.
For every A ∈ 2X, the inverse of A is the complement of A.
- D.
For every A ∈ 2X, the inverse of A is A.
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Correct answer: A, D
Key facts: symmetric difference Δ on the power set 2^X behaves like addition mod 2 on characteristic functions.
Closure: If A and B are subsets of X, then A Δ B is also a subset of X.
Associativity and commutativity: Symmetric difference is associative and commutative, so (2^X, Δ) is at least a commutative semigroup.
Identity: The empty set ∅ satisfies A Δ ∅ = A for every A ⊆ X.
Inverses: For every A ⊆ X, A Δ A = ∅, so each element is its own inverse.
Conclusion: (2^X, Δ) is an abelian group because all group axioms hold. In particular, the inverse of any A is A itself.
The statement 'H is a group.' is true.
The statement 'Every element in H has an inverse, but H is NOT a group.' is false: every element does have an inverse, but that fact together with the other axioms makes H a group.
The statement 'For every A ∈ 2^X, the inverse of A is the complement of A.' is false because A Δ A^c = X (not ∅) unless X is empty.
The statement 'For every A ∈ 2^X, the inverse of A is A.' is true because A Δ A = ∅.