GATE CS 2024 Set 1 — Question 52
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Engineering Mathematics → Analytical Aptitude → Groups, Rings & Fields
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Question
Consider the operators and defined by , , for positive integers. Which of the following statements is/are TRUE?
Correct answer
(B) Operator obeys the associative law; (D) Operator over the operator obeys the distributive law
Solution
Let us analyze each statement:Statement (A): Associativity of
Check if .
LHS:
RHS:
Since generally, is NOT associative.Statement (B): Associativity of
(standard multiplication).
Multiplication of integers is associative: .
Thus, is associative.Statement (C): Distributivity of over
Check if .
LHS:
RHS:
LHS RHS generally. Thus, does NOT distribute over .Statement (D): Distributivity of over
Check if .
LHS:
RHS:
LHS = RHS. Thus, distributes over .Therefore, statements (B) and (D) are TRUE.
Check if .
LHS:
RHS:
Since generally, is NOT associative.Statement (B): Associativity of
(standard multiplication).
Multiplication of integers is associative: .
Thus, is associative.Statement (C): Distributivity of over
Check if .
LHS:
RHS:
LHS RHS generally. Thus, does NOT distribute over .Statement (D): Distributivity of over
Check if .
LHS:
RHS:
LHS = RHS. Thus, distributes over .Therefore, statements (B) and (D) are TRUE.
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