UGC NET Questions (Paper – 1)

A gives the defining properties: reflexive, symmetric and transitive. C (equality) and D (congruence modulo n) are classic examples satisfying these properties. B is false because reflexivity cannot be dropped. E is false because equivalence classes can contain multiple elements (e.g., congruence classes modulo n). Therefore A, C and D only are correct.

Option A:
Option A is correct since it includes the definition and two standard examples while excluding B and E, which are false.

Option B:
Option B is incomplete because it ignores D, leaving out an important arithmetic example.

Option C:
Option C is wrong because it includes E, which incorrectly claims all equivalence classes are singletons.

Option D:
Option D is incorrect because it omits A, which states the defining criteria of an equivalence relation.

An equivalence relation is defined as a relation on a set that is reflexive, symmetric and transitive. Reflexive means every element is related to itself, symmetric means the relation is two-way and transitive means relations chain consistently. When all three properties hold, the relation partitions the set into equivalence classes. Therefore, a relation with these three properties is called an equivalence relation.

Option A:
A partial order is a relation that is reflexive, antisymmetric and transitive, not symmetric. It is used to model ordered structures rather than equivalence. Since the question explicitly includes symmetry, partial order cannot be the correct term.

Option B:
A symmetric relation only satisfies the symmetry property and may or may not be reflexive or transitive. The question, however, demands all three properties simultaneously. Thus, describing the relation as merely symmetric is incomplete and incorrect.

Option C:
A transitive relation satisfies just the transitivity property and may fail to be reflexive or symmetric. Again, this does not capture the full set of conditions mentioned in the question. Therefore, transitive alone is not the correct description.

Option D:
Equivalence is correct because it encapsulates the combined requirements of reflexivity, symmetry and transitivity. Such relations are crucial in classifying objects into groups where elements are "equivalent" under some criterion. Recognising this definition is important in topics like modular arithmetic and partitioning sets.