Subsets and proper subsets facts – UGC NET Question

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.