MathDB

Let

X_1, X_2, \ldots, X_{100}

be a sequence of mutually distinct nonempty subsets of a set

S

. Any two sets

X_i

and

X_{i+1}

are disjoint and their union is not the whole set

S

, that is,

X_i\cap X_{i+1}=\emptyset

and

X_i\cup X_{i+1}\neq S

, for all

i\in\{1, \ldots, 99\}

. Find the smallest possible number of elements in

S

3