Number of Unordered Pairs of Disjoint Subsets of S

To find the total number of unordered pairs of disjoint subsets of set S={1,2,3,4}, we need to consider all the possible combinations of disjoint subsets.

Understanding Disjoint Subsets
Disjoint subsets are subsets that do not have any common elements. In other words, if two subsets are disjoint, they cannot share any elements.

Finding the Disjoint Subsets
To find the disjoint subsets of a set, we can consider each element and decide whether to include it or exclude it in the subsets. Since the subsets need to be disjoint, an element can only be included in one of the subsets.

Step 1: Empty Subset
In this case, an empty subset is considered disjoint from any other subset. So, we start by including an empty subset in the pairs.

Step 2: Single Element Subsets
Next, we consider each element of the set individually and create subsets consisting of only that element. For set S={1,2,3,4}, the single-element subsets are {1}, {2}, {3}, and {4}. These subsets are also disjoint from each other.

Step 3: Two-Element Subsets
Now, we consider all possible combinations of two elements from the set. For set S={1,2,3,4}, the two-element subsets are {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, and {3,4}. These subsets are disjoint from each other and from the subsets created in the previous steps.

Step 4: Three-Element Subsets
Similarly, we consider all possible combinations of three elements from the set. For set S={1,2,3,4}, the three-element subsets are {1,2,3}, {1,2,4}, {1,3,4}, and {2,3,4}. These subsets are disjoint from each other and from the subsets created in the previous steps.

Step 5: Four-Element Subset
Finally, we consider the entire set S={1,2,3,4} as a four-element subset. This subset is disjoint from all the subsets created in the previous steps.

Total Number of Unordered Pairs
To find the total number of unordered pairs, we consider each possible combination of two subsets, one from each step. Since the subsets are disjoint, there are no restrictions on the pairs.

The total number of subsets created in the previous steps is:
- Empty subset: 1
- Single-element subsets: 4
- Two-element subsets: 6
- Three-element subsets: 4
- Four-element subset: 1

To find the total number of unordered pairs, we multiply the number of subsets from each step:
1 x (4+6+4+1) = 1 x 15 = 15

Therefore, the total number of unordered pairs of disjoint subsets of S={1,2,3,4} is 15.