How many distinct pairs of disjoint non-empty subsets of a are there, the union of which is all of a? a {0,4,5,7}

Let set A be written as B∪C, where B and C are 2 disjoint nonempty subsets of A.

the number of elements of B and C respectively can be:

i) 1, 3
ii)2, 2

for case i we have the following pairs of sets:

{0}, {4,5,7}

{4}, {0,5,7}

{5}, {0,4,5}

{7}, {0,4,5}

for case ii, consider only one of the sets, which can be any of these:

{0,4}, {0,5}, {0,7}, {4,5}, {4,7},{5,7}

Clearly these pairs complement each other, that is if the first set is  {0,4}, the second set is {5,7}, if the first set is {0,5} the second set is {4,7}, and if the first set is {0,7} the second is {4,5}.

Thus there are 4+3=7 pairs of subsets

7

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