We are asked to find the number of ways to choose two subsets \( P \) and \( Q \) from a set \( A \) of \( n \) elements, such that \( P \cap Q = \emptyset \), meaning that \( P \) and \( Q \) are disjoint.
Step 1: Understanding the choices for each element of \( A \)
Each element of \( A \) has three possibilities: 1. It can be in \( P \). 2. It can be in \( Q \). 3. It can be in neither \( P \) nor \( Q \). Since \( P \) and \( Q \) are disjoint, no element can be in both \( P \) and \( Q \) simultaneously.
Step 2: Counting the total number of ways
For each of the \( n \) elements of \( A \), there are 3 choices (either in \( P \), in \( Q \), or in neither). Therefore, the total number of ways to choose \( P \) and \( Q \) is: \[ 3^n \]
Step 3: Conclusion
Thus, the number of ways to choose \( P \) and \( Q \) such that \( P \cap Q = \emptyset \) is \( 3^n \).
\[ \boxed{3^n} \]