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Transitive subsets

Description

Let \(V_0 = \varnothing\) be the empty set and for \(i>0\), let \(V_i = \mathcal{P}(V_{i-1})\), where \(\mathcal{P}(X)\) is the set of all subsets of \(X\) (also known as the power set).  So:

\(V_1=\{\varnothing\}\)

\(V_2 = \{\varnothing,\{\varnothing\}\}\)

\(V_3=\{\varnothing,\{\varnothing\},\{\{\varnothing\}\},\{\varnothing,\{\varnothing\}\}\}\)

and so on.

A set \(X\) is called transitive if each element of \(X\) is also a subset of \(X\).


Question

How many elements in \(V_5\) are transitive?


How many elements in \(V_5\) are transitive?