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Q: let S={1,2,3,4} The total number of unordered pairs of disjoint subsets of S is equal to (A)26 (B)34 (C)42 (D)41 pls explain what the phrase"unordered pairs of disjoint subsets mean"?

Q: let S={1,2,3,4} The total number of unordered pairs of disjoint subsets of S is equal to
(A)26 (B)34 (C)42 (D)41

pls explain what the phrase"unordered pairs of disjoint subsets mean"?

Grade:11

1 Answers

bharat bajaj IIT Delhi
askIITians Faculty 122 Points
9 years ago
We first count the number of ordered pairs of disjoint subsets of S. For each element of S, we can choose to put it in either set A, set B, or neither (but not both), so we are making a sequence of 4 elements with 3 options each time. There are thus 3^4 different sequences of
choices, and each results in a distinct ordered pair of subsets.
As it is unordered pair- so we dont have to worry about the order. Hence,
If A and B are distinct, there are exactly two ordered pairs, (A,B) and (B,A), for
each unordered pair, otherwise there is just one. Given that A and B are disjoint
the only case where they are identical is when they are both empty. Thus there
are (3^4-1)/2 + 1 unordered pairs of disjoint subsets of S.
The answer = 41
Thanks
Bharat Bajaj
IIT Delhi
askiitians faculty

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