# GATE | Gate IT 2005 | Question 33

Let A be a set with n elements. Let C be a collection of distinct subsets of A such that for any two subsets S_{1} and S_{2} in C, either S_{1 }⊂ S_{2} or S_{2}⊂ S_{1}. What is the maximum cardinality of C?

**(A)** n

**(B)** n + 1

**(C)** 2^{(n-1)} + 1

**(D)** n!

**Answer:** **(B)** **Explanation:** Here let n=2 A = {1, 2}

All subsets formed by A are: – {}, {1}, {2}, {1,2}.

C is a collection of distinct subsets such that for any S1, S2 either S1⊂S2 or S2⊂S1.

So for C, {} null set can be included always since it null. set is a subset of every set.

We can choose one from either {1} or {2}, {1,2} can be included to maximise the cardinality.

So, here 1) If {1} is chosen then C = {}, {1}, {1,2} here every set is subset of other.

2) If {2} is chosen then C = {}, {2}, {1,2} here also every set is subset of other.

So, answer should be 2 but it includes empty set also therefore the maximum cardinality of C is 3.

This solution is contributed by **Anil Saikrishna Devarasetty**.

**Alternative approach –**

The description of set C in the question actually means that C is a total ordered set, so every subset of A in C should be of different size, since |A| = n, along with empty set, there are n + 1 possible sets in C, so the maximum cardinality of C is n + 1.

This solution is contributed by **Zhang Xichu**.

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