Let X={1,2,3,4,5). The number of different ordered pairs (Y,Z) that can formed such that `Y sube X,Z sube X` and `Y nn Z` is empty is

Options

A. `5^2`
B. `3^5`
C. `2^5`
D. `5^3`

Answer

To solve the problem, we need to determine the number of different ordered pairs (Y, Z) such that Y and Z are subsets of the set X = {1, 2, 3, 4, 5} and the intersection of Y and Z is empty (Y ∩ Z = ∅).

### Step-by-Step Solution:

1. **Understanding the Sets**:
We have the set X = {1, 2, 3, 4, 5}. We need to form subsets Y and Z from this set such that they do not share any elements.

2. **Considering Each Element**:
For each element in the set X, we have three choices:
- The element can be included in subset Y.
- The element can be included in subset Z.
- The element can be included in neither subset Y nor subset Z.

Note that we cannot include the element in both subsets Y and Z because their intersection must be empty.

3. **Counting Choices for Each Element**:
Since there are 3 choices for each of the 5 elements in set X, we can calculate the total number of combinations.

4. **Calculating Total Combinations**:
The total number of ordered pairs (Y, Z) can be calculated as:
\[ \text{Total combinations} = 3^5 \]
This is because each of the 5 elements has 3 independent choices.

5. **Final Calculation**:
Now, we compute \(3^5\):
\[ 3^5 = 243 \]

Thus, the number of different ordered pairs (Y, Z) such that Y ∩ Z = ∅ is **243**.

### Final Answer:
The number of different ordered pairs (Y, Z) is **243**.