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Set Theory

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We have a set of cardinality 7. Suppose P is the power set of S. We have a subset of Power set, Ps such that every pair of elements (a, b) in subset is such that, either a is a subset of b or b is a subset of a. What could be the max cardinality of Ps?
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Let $S = \{1,2,..., n\}$ and $P(S)$ be the power set of $S$.

 To maximize the property $a\subseteq b$ or $b\subseteq a$, where $a,b \in$ some subset of $P(S)$, atleast the $2$ obvious sets should be included in this i.e $\phi$ and $S$, corresponding to sets of size $0$ and $n$ respectively.

Now lets pick a set of size $1$ like $\{1\}$. Clearly, the next step is to pick a set of size $2$ which includes this chosen set of size $1$, lets say  $\{1,2\}$. Now repeat this step by considering the set $\{1,2\}$.

Generalizing this, we can say that for step $i$ we need to pick a set $X_{i} \in P(S)$ of size $i$ such that $$X_{i-1} \subseteq X_{i}\,\,\,where \,\,\,i=1...n-1$$. This totals for $n-1$ steps and in each step we choose $1$ set. Adding all of the sets we get $$2+n-1=n+1$$ as the maximum possible cardinality for such a set.
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Let us assume that we have a set of cardinality 3 and elemets are S = {1,2,3}

The power Set of this is = {Φ, {1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Now if we take any subset of this power set such that any pair (a,b) is such that a ⊆ b or b ⊆ a then for such subset we have below example:

Ps= {Φ,{1},{1,2},{1,2,3}}

as you can see that  pi and {1} are subset as pi is subset of all.

similarly {1} and {1,2} the subset property follows.

Hence for max cardinality for a set of 3, such type of subset possible for max cardinality i.e. 4.

Hence ans is n+1 = 7+1 = 8
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