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 S₁ and S₂ in C, either S₁ ⊂ S₂ or S₂⊂ S₁. 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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