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GATE 2023 CS – Question 51

Engineering Mathematics · Discrete Mathematics: Monoids and Groups · 2 marks · Multiple select

Let X be a set and $2^X$ denote the powerset of X. Define a binary operation $\Delta$ on $2^X$ as follows: $A\Delta B = (A - B) \cup (B - A)$. Let $H = (2^X, \Delta)$. 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 $A \in 2^X$, the inverse of A is the complement of A.
  4. For every $A \in 2^X$, the inverse of A is A.

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Show answer and explanation

Correct answer: (A) H is a group.; (D) For every $A \in 2^X$, the inverse of A is A.

Explanation

Symmetric difference is closed and associative, with identity ∅. A Δ A = ∅, so every element is its own inverse, and H is a group.