Write the Boolean expression for a four-input NAND gate and apply DeMorgan theorem. Draw the circuit diagram of four-input NAND gate using two input NAND gates. You may need additional logic gates to complete the diagram.
Write the Boolean expression for a four-input NAND gate and apply DeMorgan theorem. Draw the circuit diagram of four-input NAND gate using two input NAND gates. You may need additional logic gates to complete the diagram.
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
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![**Title: Understanding NAND Gates and Boolean Expressions**
**Objective:**
To explore the Boolean expression for a four-input NAND gate and apply DeMorgan's theorem. Additionally, to draw the circuit diagram of a four-input NAND gate using two-input NAND gates, possibly incorporating additional logic gates.
---
**Boolean Expression for a Four-Input NAND Gate:**
To derive the Boolean expression for a four-input NAND gate, consider the inputs as A, B, C, and D. The output \( Y \) of a four-input NAND gate can be expressed as:
\[ Y = \overline{A \cdot B \cdot C \cdot D} \]
This expression indicates that the output is the negation of the AND operation across all inputs. The NAND gate will output true if any one or more of the inputs are false.
---
**Applying DeMorgan's Theorem:**
DeMorgan's Theorem aids in simplifying expressions and is crucial for converting between logic gate implementations. For the above expression:
\[ \overline{A \cdot B \cdot C \cdot D} = \overline{A} + \overline{B} + \overline{C} + \overline{D} \]
This equivalent expression shows that a four-input NAND gate can be represented as a NOR operation between the negations of each input.
---
**Circuit Diagram Using Two-Input NAND Gates:**
To construct a four-input NAND gate using only two-input NAND gates, the following steps can be followed:
1. **Stage 1: Pairing Inputs**
- Use two NAND gates to pair the inputs:
- First NAND gate pairs inputs A and B: \( NAB = \overline{A \cdot B} \)
- Second NAND gate pairs inputs C and D: \( NCD = \overline{C \cdot D} \)
2. **Stage 2: Final NAND Operation**
- Use a third NAND gate to combine the outputs from Stage 1:
- \( Y = \overline{NAB \cdot NCD} = \overline{(\overline{A \cdot B}) \cdot (\overline{C \cdot D})} \)
This setup realizes the four-input NAND functionality using multiple two-input NAND gates.
By understanding this method, students can apply these concepts in designing more complex digital circuits.
---
This educational content provides an insightful](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf1ef7c6-a145-4dbc-a50b-84dc7d2e578d%2Fb36fef7f-8cc4-413d-a100-174f867e8a3a%2Flpe1utq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Understanding NAND Gates and Boolean Expressions**
**Objective:**
To explore the Boolean expression for a four-input NAND gate and apply DeMorgan's theorem. Additionally, to draw the circuit diagram of a four-input NAND gate using two-input NAND gates, possibly incorporating additional logic gates.
---
**Boolean Expression for a Four-Input NAND Gate:**
To derive the Boolean expression for a four-input NAND gate, consider the inputs as A, B, C, and D. The output \( Y \) of a four-input NAND gate can be expressed as:
\[ Y = \overline{A \cdot B \cdot C \cdot D} \]
This expression indicates that the output is the negation of the AND operation across all inputs. The NAND gate will output true if any one or more of the inputs are false.
---
**Applying DeMorgan's Theorem:**
DeMorgan's Theorem aids in simplifying expressions and is crucial for converting between logic gate implementations. For the above expression:
\[ \overline{A \cdot B \cdot C \cdot D} = \overline{A} + \overline{B} + \overline{C} + \overline{D} \]
This equivalent expression shows that a four-input NAND gate can be represented as a NOR operation between the negations of each input.
---
**Circuit Diagram Using Two-Input NAND Gates:**
To construct a four-input NAND gate using only two-input NAND gates, the following steps can be followed:
1. **Stage 1: Pairing Inputs**
- Use two NAND gates to pair the inputs:
- First NAND gate pairs inputs A and B: \( NAB = \overline{A \cdot B} \)
- Second NAND gate pairs inputs C and D: \( NCD = \overline{C \cdot D} \)
2. **Stage 2: Final NAND Operation**
- Use a third NAND gate to combine the outputs from Stage 1:
- \( Y = \overline{NAB \cdot NCD} = \overline{(\overline{A \cdot B}) \cdot (\overline{C \cdot D})} \)
This setup realizes the four-input NAND functionality using multiple two-input NAND gates.
By understanding this method, students can apply these concepts in designing more complex digital circuits.
---
This educational content provides an insightful
Expert Solution
Step 1: Introduce concept
Here, the task mentioned in the question is to write the expression for a four-input NAND gate.
NAND Gate
- NAND stands for NOT AND. That is its operation is inverse of AND gate.
- In AND gate, the output will be 1 (high) if and only if all of its inputs are 1 (high).
- NAND is reverse of AND. It means it will produce output as 1 in all conditions except when all of its inputs are 1.
- NAND gate is a universal gate as it can be used to make all other logic gates.
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