Hi there! Thank you for the opportunity to help you with your question!

This is a problem involving combinations. We want pairs of disjoint, non-empy subsets, whose union is A.

An example of such pair would be {0} and {2,6,8,9}. But we could just as well have chosen {2} and {0, 6,8,9}, or even {0, 2} and {6, 8, 9}. Listing all the combinations in this case is actually doable. But there is a slightly easier way, we just have to be careful.

There are multiple ways to make pairs, but the total number of elements in the union of both sets must be equal to the number of elements in A, or 5. The possible ways to make 5 consist of either: a set with 1 element and another with 4; Or one set with 2 elements and another with 3.

In the first case, there are 5 possible one set elements. (5 choose 1).

In the second case, there are 10 possible two set elements (5 choose 2).

Therefore, there are 15 total pairs

Just as an exercise, because I have extra time, let's list them all:

{0}, {2,6,8,9}

{2}, {0,6,8,9}

{6}, {0,2,8,9}

{8}, {0,2,6,9}

{9}, {0,2,6,8}

{0,2}{6,8,9}

{0,6}{2,8,9}

{0,8}{2,6,9}

{0,9}{2,6,8}

{2,6}{0,8,9}

{2,8}{0,6,9}

{2,9}{0,6,8}

{6,8}{0,2,9}

{6,9}{0,2,8}

{8,9}{0,2,6}

(If you try to make any other pairs, they will just be a permutation (that is a reordering). Since sets don't have order, this new pair would be equivalent to one of the above pairs!)

Please let me know if you need any clarification. Always glad to help!

Sep 1st, 2015

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