Binary operation algebraic properties – UGC NET Question

Statements A and B give the definition of a binary operation and associativity. D is also correct since there are operations that are associative but not commutative, such as matrix multiplication. C is false because commutativity must hold for all pairs, not just some, and E is false because commutativity does not imply associativity. Thus A, B and D only are correct.

Option A:
Option A is incomplete as it omits D, missing the important fact that associativity and commutativity are independent properties.

Option B:
Option B is wrong because it includes C, which weakens commutativity incorrectly by requiring it only for a single pair.

Option C:
Option C is incorrect since it omits A, the definition of a binary operation, which is one of the correct statements.

Option D:
Option D is correct because it gathers the proper definition, the associativity law and a fact about independence of associativity from commutativity, while excluding C and E, which are incorrect.