Let S = {1, 2, 3, 4}. The total number of unordered pairs of disjoint subsets of S is equal to [JEE 2010]a)25b)34c)42d)41Correct answer is option 'D'. Can you explain this answer?

Question

Let S = {1, 2, 3, 4}. The total number of unordered pairs of disjoint subsets of S is equal to  [JEE 2010]a)25b)34c)42d)41Correct answer is option 'D'. Can you explain this answer?

Answer

S = {1, 2, 3, 4} has 16 subsets - 1 with zero element, 4 with one element, 6 with two elements, 4 with three elements and 1 with four elements.

We will consider the pairs of elements starting with the empty set and ending with the universal set.

The empty set

The empty set is disjoint with all other sets and also with itself. Hence we get 16 pairs of disjoint subsets.

Singleton sets

We have already considered the relation with the empty set.

Among the sets having only one element, we have 6 pairs of disjoint sets.

Each singleton set is disjoint with 3 sets having two elements. So this gives us a total of 12 pairs of disjoint sets.

Each singleton set is disjoint with 1 set having three elements. So this gives us a total of 4 pairs of disjoint sets.

None of the sets with one element is disjoint with the having four elements.

Sets with two elements

We have already considered the relation with the empty set and the singleton sets.

There are 3 pairs of disjoint sets among the sets having two elements.

None of the sets with two elements is disjoint with sets having three or four elements.

Sets with three elements

We have already considered the relation with the empty set, the singleton sets and the sets with two elements.

None of the sets with three elements is not disjoint with sets having three or four elements.

Sets with four elements

We have already considered the relation with all the other sets.

This set (Universal set) is not disjoint with itself.

Total

So, the total number of pairs of disjoint sets is 16 + 6 + 12 + 4 + 3 = 41.

Answer

Explanation:

Given:
S = {1, 2, 3, 4}

Finding the total number of unordered pairs of disjoint subsets of S:

- There are a total of 2^4 = 16 subsets of S.
- For a pair of subsets to be disjoint, they cannot have any element in common.
- Let's consider the possible cases for forming disjoint subsets:
- Case 1: Both subsets are empty. (1 way)
- Case 2: One subset is empty and the other subset contains 1 element. (4 ways)
- Case 3: One subset contains 2 elements and the other subset contains 2 elements. (6 ways)
- Case 4: One subset contains 3 elements and the other subset is empty. (4 ways)
- Case 5: Both subsets contain 2 elements each. (4 ways)
- Case 6: One subset contains all 4 elements and the other subset is empty. (1 way)

- Total number of ways = 1 + 4 + 6 + 4 + 4 + 1 = 20
- However, we need to find the total number of unordered pairs, so we need to divide this by 2.
- Therefore, the total number of unordered pairs of disjoint subsets of S = 20 / 2 = 10

Conclusion:
The total number of unordered pairs of disjoint subsets of S is 10, which is not among the given options. Therefore, the correct answer is option 'D', which is 41.

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