Question

let S = {1,2,3,4}. The total no. of unordered pairs of disjoint subsets of S is equal to

Answer 1

HEY MATE HERE IS YOUR ANSWER

S=(1, 2,3,4) has 16 subsets 1 with zero element, 4 with one element, 6 with two element, 4 with three element and 1 with four element.

Hence we get 16 pairs of disjoint subsets.

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Answer 2

A pair of disjoint subsets is formed by examining each element of SS in turn and deciding whether to put it into one subset, into the other subset, or into neither.   Three choices for each of 4 elements gives us the term 3434 as the count of all possible ordered pairs of disjoint subsets that can be formed.

To count only unordered pairs, we need to determine how to group every ordered pair into equivalence classes.   Now, each pair is one of 2!2!permutations, except the pair ⟨∅,∅⟩⟨∅,∅⟩.  ( The null set is disjoint with itself, as ∅∩∅=∅∅∩∅=∅ )

Hence the count of unordered pairs of disjoint subsets of SS is : 34−12!+1