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.

A is correct because subset means every element of A belongs to B. D is true as the empty set, having no elements, vacuously satisfies the subset condition for any set. E is also correct because mutual inclusion implies equality of sets. B is false since a proper subset is strictly smaller than the set, and C is false because the universal set contains all elements, so other sets are subsets of U, not the other way round. Thus A, D and E only are correct.

Option A:
Option A lists A and D but omits E, failing to express how mutual subset relations guarantee equality. Without E, the account of subset relations is incomplete.

Option B:
Option B includes A and E but omits D, ignoring the important property of the empty set as a subset of every set. This makes the combination less comprehensive.

Option C:
Option C is correct because it combines all three true statements and excludes B and C, which misrepresent proper subset and universal set roles. It fully reflects the basic set relations used in reasoning questions.

Option D:
Option D includes only D and E, omitting A, and therefore does not explicitly state what it means for A to be a subset of B. This leaves out a fundamental definition.