Cardinality and Repeated Elements

The notation n(S) represents the cardinality of S, meaning the number of distinct elements contained in the set. The empty set, denoted by ∅, contains no elements at all. Therefore its cardinality is n(S) = 0. Option B would be correct for a singleton set, which has exactly one element, not for ∅. Option C gives the empty-set symbol rather than a numerical cardinality, and option D is incorrect because the cardinality of an empty set is well defined. A useful distinction is that ∅ has zero elements, whereas {∅} has one element—the empty set itself. Thus option A is correct.

The governing concept is the cardinality of a set. The notation n(Y) means the number of distinct elements contained in set Y, not the greatest element, the sum of the elements, or the value of a pattern. The roster form of Y lists 2, 4, 6, 8, and 10. Each number is different, so counting the members gives five elements: n(Y) = 5. Therefore option A is correct. Option B counts only four members and misses one listed value. Option C is the largest element, 10, but the largest element is not the cardinality. Option D is the sum, because 2 + 4 + 6 + 8 + 10 = 30; addition produces a different quantity. Since there are no repeated entries to remove, the final cardinality remains 5.