Question:
Let \( G \) be the power set of \( \{1, 2, 3, 4\} \). Let the binary operation \( \Delta \) on \( G \) be
Let \( G \) be the power set of \( \{1, 2, 3, 4\} \). Let the binary operation \( \Delta \) on \( G \) be
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When working with sets and binary operations, the identity element and the order of elements are key properties that define the structure of the group.
Updated On: Jun 1, 2026
- \( \{1\} \) is an identity element of \( (G, \Delta) \).
- \( (G, \Delta) \) is an abelian group but not cyclic.
- \( (G, \Delta) \) is a group and has an element of order 4.
- \( (G, \Delta) \) is a group and has an element of order 8.
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The Correct Option is B
Solution and Explanation
Step 1: Check the properties of \( (G, \Delta) \).
We need to verify if the structure \( (G, \Delta) \) satisfies the group properties. The set \( G \) consists of the power set of \( \{1, 2, 3, 4\} \), which includes all subsets of this set, and \( \Delta \) is the symmetric difference between sets.
The operation \( \Delta \) is associative, and there exists an identity element \( \{1\} \), where the symmetric difference of any set with \( \{1\} \) returns the set itself.
Step 2: Identity element check.
The identity element in a group is an element that, when combined with any other element of the group, leaves the other element unchanged.
The identity element in \( (G, \Delta) \) is \( \{1\} \), as \( A \Delta \{1\} = A \) for any subset \( A \). Therefore, statement (A) is true.
Step 3: Abelian check.
The operation \( \Delta \) is commutative, meaning that for any two sets \( A \) and \( B \), \( A \Delta B = B \Delta A \). Therefore, \( (G, \Delta) \) is an abelian group.
Step 4: Group with element of order 4.
The group \( (G, \Delta) \) is indeed a group and contains an element of order 4. The order of an element is the smallest number of applications of the operation required to return the identity element.
In this case, there are elements that satisfy this condition, so statement (C) is correct.
Step 5: Conclusion.
Based on the analysis, statement (C) is the true statement, as \( (G, \Delta) \) is a group and has an element of order 4. Therefore, the correct answer is (C).
We need to verify if the structure \( (G, \Delta) \) satisfies the group properties. The set \( G \) consists of the power set of \( \{1, 2, 3, 4\} \), which includes all subsets of this set, and \( \Delta \) is the symmetric difference between sets.
The operation \( \Delta \) is associative, and there exists an identity element \( \{1\} \), where the symmetric difference of any set with \( \{1\} \) returns the set itself.
Step 2: Identity element check.
The identity element in a group is an element that, when combined with any other element of the group, leaves the other element unchanged.
The identity element in \( (G, \Delta) \) is \( \{1\} \), as \( A \Delta \{1\} = A \) for any subset \( A \). Therefore, statement (A) is true.
Step 3: Abelian check.
The operation \( \Delta \) is commutative, meaning that for any two sets \( A \) and \( B \), \( A \Delta B = B \Delta A \). Therefore, \( (G, \Delta) \) is an abelian group.
Step 4: Group with element of order 4.
The group \( (G, \Delta) \) is indeed a group and contains an element of order 4. The order of an element is the smallest number of applications of the operation required to return the identity element.
In this case, there are elements that satisfy this condition, so statement (C) is correct.
Step 5: Conclusion.
Based on the analysis, statement (C) is the true statement, as \( (G, \Delta) \) is a group and has an element of order 4. Therefore, the correct answer is (C).
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