The logic circuit shown in the diagram directly implements what Boolean expression? Show intermediate values, too. y
The logic circuit shown in the diagram directly implements what Boolean expression? Show intermediate values, too. y
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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![### Educational Website Content
#### Understanding Boolean Expressions Through Logic Circuits
**Question:**
The logic circuit shown in the diagram directly implements what Boolean expression? Show intermediate values, too.
**Diagram Overview:**
The logic circuit provided is a combination of various logic gates: AND, OR, and NOT gates. Let's break it down step-by-step to establish the complete Boolean expression and the intermediate values.
#### Step-by-Step Breakdown
1. **Inputs:**
- \( x \)
- \( y \)
- \( z \)
2. **Gates and Intermediate Values:**
- **NOT Gate:** The input \( x \) is fed into a NOT gate.
- Output of NOT gate: \( \overline{x} \)
- **First AND Gate:**
- Inputs: \( \overline{x} \) (output from the NOT gate), and \( y \)
- Output: \( \overline{x} \cdot y \)
- **Second AND Gate:**
- Inputs: \( y \) and \( z \)
- Output: \( y \cdot z \)
- **OR Gate (for combining above AND gates' outputs):**
- Inputs: \( \overline{x} \cdot y \) and \( y \cdot z \)
- Output: \( (\overline{x} \cdot y) + (y \cdot z) \)
- **Final AND Gate:**
- Inputs: \( x \) and the output from the previous OR gate \( (\overline{x} \cdot y) + (y \cdot z) \)
- Output: \( x \cdot [(\overline{x} \cdot y) + (y \cdot z)] \)
3. **Final Boolean Expression:**
- The overall Boolean expression implemented by this circuit is:
\[
F = x \cdot \left[(\overline{x} \cdot y) + (y \cdot z)\right]
\]
This completes our analysis of the logic circuit diagram. The final Boolean expression illustrates how the circuit processes the inputs to produce the desired output.
By understanding each intermediate step and value, we can gain a deeper insight into how complex logic circuits decompose into simpler Boolean algebraic expressions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4ad255c4-ef9f-4006-8c97-7f5da0743c94%2F66c62514-aeeb-4dc3-82ee-adb4c41f9f9a%2F4n5tbx9_processed.png&w=3840&q=75)
Transcribed Image Text:### Educational Website Content
#### Understanding Boolean Expressions Through Logic Circuits
**Question:**
The logic circuit shown in the diagram directly implements what Boolean expression? Show intermediate values, too.
**Diagram Overview:**
The logic circuit provided is a combination of various logic gates: AND, OR, and NOT gates. Let's break it down step-by-step to establish the complete Boolean expression and the intermediate values.
#### Step-by-Step Breakdown
1. **Inputs:**
- \( x \)
- \( y \)
- \( z \)
2. **Gates and Intermediate Values:**
- **NOT Gate:** The input \( x \) is fed into a NOT gate.
- Output of NOT gate: \( \overline{x} \)
- **First AND Gate:**
- Inputs: \( \overline{x} \) (output from the NOT gate), and \( y \)
- Output: \( \overline{x} \cdot y \)
- **Second AND Gate:**
- Inputs: \( y \) and \( z \)
- Output: \( y \cdot z \)
- **OR Gate (for combining above AND gates' outputs):**
- Inputs: \( \overline{x} \cdot y \) and \( y \cdot z \)
- Output: \( (\overline{x} \cdot y) + (y \cdot z) \)
- **Final AND Gate:**
- Inputs: \( x \) and the output from the previous OR gate \( (\overline{x} \cdot y) + (y \cdot z) \)
- Output: \( x \cdot [(\overline{x} \cdot y) + (y \cdot z)] \)
3. **Final Boolean Expression:**
- The overall Boolean expression implemented by this circuit is:
\[
F = x \cdot \left[(\overline{x} \cdot y) + (y \cdot z)\right]
\]
This completes our analysis of the logic circuit diagram. The final Boolean expression illustrates how the circuit processes the inputs to produce the desired output.
By understanding each intermediate step and value, we can gain a deeper insight into how complex logic circuits decompose into simpler Boolean algebraic expressions.
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