UGC NET Questions (Paper – 1)

Statements A and B are standard properties: a set includes itself as a subset and ∅ is a subset of every set. C is correct since a proper subset of a finite set must omit at least one element, so its cardinality is smaller. E is also correct because mutual inclusion implies equality of sets. D is false; no set is a proper subset of itself by definition. Thus A, B, C and E only are correct.

Option A:
Option A is incomplete because it omits E and therefore fails to mention the important characterisation of equality in terms of mutual subset relations, a common exam point.

Option B:
Option B is wrong since it adds D, which incorrectly states that a set can be a proper subset of itself, contradicting the strictness in the definition of “proper”.

Option C:
Option C is incorrect because it includes D and drops C, losing the crucial cardinality property of proper subsets and introducing the false claim that a set is a proper subset of itself.

Option D:
Option D is correct as it collects all true subset properties listed and excludes D, which conflicts with the standard definition of proper subset in set theory.

Statements A, B and C give standard definitions of injective, surjective and bijective functions. E is also correct in the finite equal-cardinality case: an injective map from A to B must be surjective. D is false because surjectivity alone does not guarantee injectivity; many onto but not one-to-one functions exist. Therefore A, B, C and E only are correct.

Option A:
Option A is correct as it aggregates all the true statements about these function types, including the finite-set result, while leaving out D, which wrongly treats surjectivity as forcing injectivity.

Option B:
Option B is incorrect because it omits E, and E is a true theorem for finite sets of equal size (injective ⇒ surjective).

Option C:
Option C is incorrect because it omits A (a true definition of injective), so it does not list all correct statements.

Option D:
Option D is incorrect because it omits B (a true definition of surjective), so it does not list all correct statements.