Final answer:
The cardinality of the power set (∣P(S)∣) represents the number of subsets of a set S, including the empty set and S itself. The cardinality of P(S) is calculated as 2^n, where n is the size of set S. So, it is possible for ∣P(S)∣ to equal 10, but it depends on the specific size of set S. So, the correxct option is c) This can happen, but it doesn't always happen.
Explanation:
The cardinality of the power set (∣P(S)∣) is determined by the number of subsets a set S can have, considering the inclusion of the empty set and the set itself. Mathematically, ∣P(S)∣ is calculated as 2^n, where n represents the size of set S. In this context, the statement ∣P(S)∣=10 is plausible, but not universally applicable. The actual cardinality depends on the specific size and elements of set S.
Option (c) "This can happen, but it doesn't always happen" is the correct choice, as the equality ∣P(S)∣=10 is contingent on the particular characteristics of S. It signifies a scenario where there exists a set S for which the power set has precisely 10 subsets, including the empty set and S itself. However, this condition is not a constant truth for all sets; the cardinality varies based on the size and content of each specific set.
Understanding the relationship between the cardinality of the power set and the size of the original set provides insight into the diverse possibilities within set theory. The flexibility of this relationship highlights the nuanced nature of mathematical concepts and the need for context-specific considerations.