Let S = {1, 2, 3, 4). The total number of unordered pairs of disjoint subsets of S is equal to?Sir please explain the solution by giving few examples, it can be lengthy answer but explain everything. And please tell me why we multiply 3 for 4 times?Please..I am confused in some basic concepts

Asked by Atul | 28th Apr, 2020, 11:17: PM

Expert Answer:

S={1,2,3,4}
The total number of sebsets = 24 = 16
Now, the unordered pairs of disjoint subsets can be obtained as follows:
1. Let's take one of the subsets in the unordered pair as {1}, then the unordered pairs of disjoint subsets are
{1}, {2}
{1}, {3}
{1}, {4}
{1}, {2, 3}
{1}, {3, 4}
{1}, {2, 4}
{1}, {2, 3, 4}
 
2. Let's take one of the subsets in the unordered pair as {2}, then the unordered pairs of disjoint subsets are
{2}, {3}
{2}, {4}
{2}, {1, 3}
{2}, {1, 4}
{2}, {3, 4}
{2}, {1, 3, 4}

3. Let's take one of the subsets in the unordered pair as {3}, then the unordered pairs of disjoint subsets are
{3}, {4}
{3}, {1, 2}
{3}, {1, 4}
{3}, {2, 4}
{3}, {1, 2, 4}
 
4. Let's take one of the subsets in the unordered pair as {4}, then the unordered pairs of disjoint subsets are
{4}, {1, 2}
{4}, {1, 3}
{4}, {2, 3}
{4}, {1, 2, 3}
 
5. Subsets with no singletons
{1, 2}, {3, 4}
{2, 3}, {1, 4}
{2, 4}, {1, 3}
 
This gives 7+6+5+4+3=25
If one of the subsets in the unordered pair is empty, the subsets will be disjoint.
This gives 16 unordered disjoint pairs.
Hence, the total number od unordered pairs of disjoint subsets of S is 25+16=41.

Answered by Renu Varma | 30th Apr, 2020, 10:57: AM

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