Let S = {1,2,3,4}. The total number of unordered pairs of disjoint subsets of S is equal to

Options

A. 34
B. 43
C. 42
D. 41

Answer

To find the total number of unordered pairs of disjoint subsets of the set \( S = \{1, 2, 3, 4\} \), we can follow these steps:

### Step 1: Understand the concept of disjoint subsets
Disjoint subsets are subsets that do not share any elements. For example, if we have subsets \( A \) and \( B \), they are disjoint if \( A \cap B = \emptyset \).

### Step 2: Determine the number of elements in set \( S \)
The set \( S \) has 4 elements: \( 1, 2, 3, 4 \). Thus, \( n = 4 \).

### Step 3: Calculate the total number of ways to assign each element
Each element in the set \( S \) can be assigned to one of three categories:
1. It can belong to subset \( A \).
2. It can belong to subset \( B \).
3. It can belong to neither subset.

Since the subsets are disjoint, an element cannot belong to both subsets at the same time.

### Step 4: Use the formula for the number of unordered pairs of disjoint subsets
The total number of unordered pairs of disjoint subsets can be calculated using the formula:
\[ \frac{3^n + 1}{2} \]
where \( n \) is the number of elements in set \( S \).

### Step 5: Substitute \( n \) into the formula
Now, substituting \( n = 4 \):
\[ \frac{3^4 + 1}{2} \]

### Step 6: Calculate \( 3^4 \)
Calculating \( 3^4 \):
\[ 3^4 = 81 \]

### Step 7: Add 1 to \( 3^4 \)
Now, add 1 to \( 81 \):
\[ 81 + 1 = 82 \]

### Step 8: Divide by 2
Finally, divide by 2:
\[ \frac{82}{2} = 41 \]

### Conclusion
Thus, the total number of unordered pairs of disjoint subsets of \( S \) is \( 41 \).

### Final Answer
The answer is \( 41 \).

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