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

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

Every element in H has an inverse, but H is NOT a group.

For every A ∈ 2^X, the inverse of A is the complement of A.

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

GATE CSE 2023 Q51: Power set under symmetric difference as a group

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?

Try it: multiple select: one or more correct options

Select all correct options. No negative marking, no partial marks.

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Answer: A, D (official GATE 2023 final answer key)

Concept used: Group axioms for symmetric difference: closure, associativity, identity $\emptyset$ , inverses.

First step: Find the identity element $E$ with $A \,\Delta\, E = A$ , then compute $A \,\Delta\, A$ .

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Source: official GATE 2023 CS question paper and final answer key, organised by IIT Kanpur. Question paper · Answer key