UGC NET Questions (Paper – 1)

Statements A and B correctly define closure and commutativity for binary operations, D notes that exams introduce custom operations to test rule-following rather than standard arithmetic and E emphasises the need to use the given definition faithfully. Statement C is false because associativity means (a * b) * c = a * (b * c), not that the results are always different. Hence, the combination A, B, D and E only is correct.

Option A:
Option A is incomplete as it omits E and thus fails to stress that candidates must not revert to usual addition or multiplication when a custom rule is specified. This omission makes the description less helpful for exam strategy.

Option B:
Option B is also incomplete because it leaves out A, so it does not explicitly mention closure, the idea that combining elements of S stays within S, which is often implied when operations are defined.

Option C:
Option C is wrong since it includes C, which reverses the meaning of associativity by saying results are always different, and omits B, thereby ignoring the commutative law. This mixture of an error and an omission makes the option invalid.

Option D:
Option D is correct because it captures closure, commutativity, exam usage of custom operations and the necessity of following the definition, while excluding C, which misstates associativity. It reflects the way NET questions check conceptual clarity about operations.

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.