GATE CS 2023 Set 1 — Question 51
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Engineering Mathematics → Sets & Combinatorics → Groups, Rings & Fields
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Question
Let be a set and denote the powerset of .
Define a binary operation on as follows:Let . Which of the following statements about is/are correct?
Define a binary operation on as follows:Let . Which of the following statements about is/are correct?
Correct answer
(A) H is a group.; (D) For every A ∈ 2^X, the inverse of A is A.
Solution
To determine if is a group, we check the group axioms for the symmetric difference operation :
Similarly, . Thus, is the identity element.
Thus, every element is its own inverse ().Evaluating the options:
1.Closure: For any , is also a subset of , so .
2.Associativity: It is a known property of symmetric difference that for all .
3.Identity Element: Let be the empty set. For any :
.Similarly, . Thus, is the identity element.
4.Inverse Element: For any , we look for such that :
.Thus, every element is its own inverse ().Evaluating the options:
- (A) is correct: satisfies all group axioms.
- (B) is incorrect: Since is a group.
- (C) is incorrect: The inverse of is itself. The symmetric difference of and its complement is . This is only the identity if .
- (D) is correct: As shown above, , so the inverse of is .
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