Minimize the following Boolean function  F (P, Q, R, S) = π (3, 5, 7, 8, 10, 11, 12, 13) to PoS form. 

To minimize the given Boolean function \( F(P, Q, R, S) = \pi(3, 5, 7, 8, 10, 11, 12, 13) \) to the Product of Sums (PoS) form, we'll follow these steps:

1. **List the minterms**: First, convert the maxterm indices to the minterms that are missing from the range (0 to 15).

2. **Group the missing minterms**: These missing minterms will help us identify the minterms that should be 1 in the Truth Table.

3. **Karnaugh Map (K-map) analysis**: Place the 0s (from the given indices) on the K-map to find simplified sum terms.

4. **Find the simplified maxterms** from the K-map to obtain the PoS expression.

### Step-by-step Solution:

1. **Convert maxterms to minterms**:
   - Maxterms given: \( \pi (3, 5, 7, 8, 10, 11, 12, 13) \)
   - Minterms are all terms from 0 to 15 not listed above, which are:  
     \( m(0, 1, 2, 4, 6, 9, 14, 15) \).

2. **Create the K-map** and place 0s for the given maxterms:
   - 2-variable map with P and Q as the row variables and R and S as the column variables.

| PQ\RS | 00 | 01 | 11 | 10 |
|-------|----|----|----|----|
| **00** | 1  | 1  | 0  | 0  |
| **01** | 1  | 0  | 0  | 1  |
| **11** | 0  | 0  | 0  | 1  |
| **10** | 0  | 1  | 1  | 0  |

3. **Identify groups of 0s and derive the maxterms** from the map. The grouping will help simplify the expression:

   - **Group 1:** Maxterms that cover columns RS = 00 → (P + R)  
   - **Group 2:** Row 01 with column RS = 11 → \( (P + S) \)  
   - **Group 3:** RS = 10, PQ = 11 and PQ = 10 → (Q + R)

**4. Simplified PoS:**
So, the minimized function in Product of Sum form is:
   
\[
F(P, Q, R, S) = (P + R)(P + S)(Q + R)
\] 

Do you need further clarifications? 

Here are some further questions for exploring the concept:
1. What is the meaning of maxterm in Boolean functions?
2. How do minterms and maxterms relate?
3. Can you create a truth table for the simplified PoS?
4. How is the grouping performed in the Karnaugh map?
5. Why PoS forms are useful in digital circuits? 

**Tip**: Always check your simplified result by creating truth tables to verify logic validity!

Minimize the following Boolean function F (P, Q, R, S) = π (3, 5, 7, 8, 10, 11, 12, 13) to PoS form.

Learn how to minimize the Boolean function F(P, Q, R, S) = π(3, 5, 7, 8, 10, 11, 12, 13) to its simplified Product of Sums (PoS) form. This tutorial covers key concepts such as Karnaugh Maps and the process of PoS simplification, ideal for Grades 11-12 and undergraduate students.

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