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 s1 s2 or s2 s1. What is the maximum cardinality of c?

Answer 1

Answer:

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.