UGC NET Questions (Paper – 1)

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.

Statements A, B and D describe core properties of common binary operations. A correctly defines a binary operation as a rule from ordered pairs in S back to S, capturing closure. B is true because addition of reals satisfies both commutativity and associativity. D is also correct since matrix multiplication, though associative, is in general not commutative. Statement C is false because subtraction is neither commutative nor associative, and E is false since having an identity element does not ensure that every element has an inverse unless additional conditions such as group structure hold. Hence the correct combination is A, B and D only.

Option A:
Option A is incorrect as it lists only A and B and omits D, which conveys an important contrast between addition and matrix multiplication. Without D, the option does not fully represent the intended range of binary operation examples. It is therefore not the best match for the correct set.

Option B:
Option B is wrong because it includes only B and D, dropping the general definition of a binary operation in A. While B and D are true, the absence of A means the option does not capture the foundational concept being tested. This makes it incomplete.

Option C:
Option C is correct since it includes the definition in A, the familiar properties of addition in B and the non-commutativity of matrix multiplication in D, while excluding C and E, both of which misstate important algebraic facts. This aligns with the structure of binary operations as tested in NET mathematics.

Option D:
Option D is incorrect because it incorporates C, which wrongly claims that subtraction is commutative, along with true statements. Once a false statement is added to the combination, it cannot represent the exact set of correct statements.