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Question

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 S1S2 or S2S1. What is the maximum cardinality of C ?

A
n
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B
n+1
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C
2n1+1
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D
n!
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Solution

The correct option is B n+1
The way C is defined in the question, it contains only comparable subsets of A.
i.e. the set C is the set of all comparable subsets of the set A. Such a set is called a chain.
Consider an example with set A={1,2}
Subsets are ϕ,{1},{2} and {1,2}
Maximum cardinality of collection of distinct subsets is {ϕ,{1} and {1,2}} i.e. 3
So with n elements maximum cardinality is n+1.

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