Consider sequence of sets defined as follows: Bn. The cardinality
Consider a sequence of n sets defined as follows: B1 ⊆ B2 ⊆ B3 ⊆ … ⊆ Bn. The cardinality of the set Bn is n. What is the maximum value of |B1| + |B2| + … + |Bn|?
- A.
n
- B.
n2
- C.
n(n+1)/2
- D.
None of these
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Correct answer: B
If the sets B1,B2,...,Bn are not strictly contained within one another (i.e., they can be equal), then indeed each set can potentially contain all elements. Here's why n^2 makes sense:
Each set can contain up to n elements.
Since we are summing the cardinalities and each of the n sets can have up to n elements, we get n×n=n^2.
Thus, if the sets can be equal and each set contains the maximum n elements, the total sum is n^2.