II. (a) Construct a Boolean expression having the circuit shown below: OR Q AND R AND the circuit in each case when input values are as follows (b) Find the output, S, of POR S 0 1 1 10 1 11 1 Case 1: Case 2: Case 3:
II. (a) Construct a Boolean expression having the circuit shown below: OR Q AND R AND the circuit in each case when input values are as follows (b) Find the output, S, of POR S 0 1 1 10 1 11 1 Case 1: Case 2: Case 3:
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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discrete structures
![### Boolean Expression and Circuit Diagram Analysis
#### Part (a): Construct a Boolean Expression
The circuit depicted consists of the following elements:
1. **OR Gate**: Takes inputs \( P \) and \( Q \).
2. **AND Gate**: Takes input \( R \) and another input derived from the OR gate's output.
3. **NOT Gate (Inverter)**: Inverts the output of the AND gate that takes \( R \) and the OR gate's output.
4. **Second AND Gate**: Combines the result of the OR gate and the inverted signal from the NOT gate to produce the final output \( S \).
To construct the Boolean expression:
- The OR gate outputs \( P + Q \).
- The first AND gate takes inputs \( (P + Q) \) and \( R \), outputting \( (P + Q)R \).
- The NOT gate inverts the result of the first AND gate, resulting in \( \overline{(P + Q)R} \).
- The second AND gate combines \( (P + Q) \) and \( \overline{(P + Q)R} \), giving the final output:
\[
S = (P + Q) \cdot \overline{(P + Q)R}
\]
#### Part (b): Determine the Output \( S \)
Evaluate the output \( S \) for each case described in the table:
- **Case 1:**
- Inputs: \( P = 0, Q = 1, R = 1 \)
- \( P + Q = 0 + 1 = 1 \)
- \( (P + Q)R = 1 \times 1 = 1 \)
- \( \overline{(P + Q)R} = \overline{1} = 0 \)
- \( S = 1 \times 0 = 0 \)
- **Case 2:**
- Inputs: \( P = 1, Q = 0, R = 1 \)
- \( P + Q = 1 + 0 = 1 \)
- \( (P + Q)R = 1 \times 1 = 1 \)
- \( \overline{(P + Q)R} = \overline{1} = 0 \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F75c1327d-7664-4a3e-b58c-1991323ef32f%2Fd54b82e3-c0a5-43c0-bffe-f0672ed24943%2Fhfo5tq_processed.png&w=3840&q=75)
Transcribed Image Text:### Boolean Expression and Circuit Diagram Analysis
#### Part (a): Construct a Boolean Expression
The circuit depicted consists of the following elements:
1. **OR Gate**: Takes inputs \( P \) and \( Q \).
2. **AND Gate**: Takes input \( R \) and another input derived from the OR gate's output.
3. **NOT Gate (Inverter)**: Inverts the output of the AND gate that takes \( R \) and the OR gate's output.
4. **Second AND Gate**: Combines the result of the OR gate and the inverted signal from the NOT gate to produce the final output \( S \).
To construct the Boolean expression:
- The OR gate outputs \( P + Q \).
- The first AND gate takes inputs \( (P + Q) \) and \( R \), outputting \( (P + Q)R \).
- The NOT gate inverts the result of the first AND gate, resulting in \( \overline{(P + Q)R} \).
- The second AND gate combines \( (P + Q) \) and \( \overline{(P + Q)R} \), giving the final output:
\[
S = (P + Q) \cdot \overline{(P + Q)R}
\]
#### Part (b): Determine the Output \( S \)
Evaluate the output \( S \) for each case described in the table:
- **Case 1:**
- Inputs: \( P = 0, Q = 1, R = 1 \)
- \( P + Q = 0 + 1 = 1 \)
- \( (P + Q)R = 1 \times 1 = 1 \)
- \( \overline{(P + Q)R} = \overline{1} = 0 \)
- \( S = 1 \times 0 = 0 \)
- **Case 2:**
- Inputs: \( P = 1, Q = 0, R = 1 \)
- \( P + Q = 1 + 0 = 1 \)
- \( (P + Q)R = 1 \times 1 = 1 \)
- \( \overline{(P + Q)R} = \overline{1} = 0 \)
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