Convert the following Boolean equation to standard SOP form and implement using NAND-NAND logic: (i) y = A'B'C' + ABC'D' + A'C'D' (ii) y = PQ + P'QR + PQR.
2 Answers
To convert Boolean equations to standard Sum of Products (SOP) form and implement them using NAND-NAND logic, follow these steps:
### 1. Boolean Equation: \( y = A'B'C' + ABC'D' + A'C'D' \)
#### Convert to Standard SOP Form
The given equation \( y = A'B'C' + ABC'D' + A'C'D' \) is already in the standard SOP form since it is a sum (OR) of products (ANDs) of literals. Each term in the sum is a product of literals, and this is the definition of SOP form.
- **Term 1**: \( A'B'C' \)
- **Term 2**: \( ABC'D' \)
- **Term 3**: \( A'C'D' \)
#### Implement using NAND-NAND Logic
To implement this equation using NAND gates, follow these steps:
1. **Write each term as a NAND expression:**
For NAND gate implementation, first express each AND and OR operation using NAND gates:
- **NAND Gate**: \( A \cdot B \) is implemented as \( \overline{\overline{A \cdot B}} \), where \( \overline{A \cdot B} \) is the NAND operation.
- **OR Gate**: \( A + B \) is implemented as \( \overline{\overline{A} \cdot \overline{B}} \).
2. **Implement the product terms:**
- **Term 1**: \( A'B'C' \)
- First, find \( A' = \overline{A} \), \( B' = \overline{B} \), and \( C' = \overline{C} \) using NAND gates: \( A' = A \text{ NAND } A \), and similarly for \( B \) and \( C \).
- Then, \( A' \cdot B' \cdot C' \) is implemented as \( \overline{\overline{A' \cdot B'} \cdot \overline{C'}} \).
- **Term 2**: \( ABC'D' \)
- Find \( C' = \overline{C} \) and \( D' = \overline{D} \).
- Then, \( A \cdot B \cdot C' \cdot D' \) is implemented as \( \overline{\overline{A \cdot B \cdot C' \cdot D'}} \).
- **Term 3**: \( A'C'D' \)
- Use \( A' = \overline{A} \), \( C' = \overline{C} \), and \( D' = \overline{D} \).
- Then, \( A' \cdot C' \cdot D' \) is implemented as \( \overline{\overline{A' \cdot C' \cdot D'}} \).
3. **Combine the terms using NAND gates:**
To sum the products, you need to OR the terms together. This can be done using NAND gates:
- Implement OR operation by combining the results of the NAND gates for the product terms.
- The OR of three terms \( A, B, \) and \( C \) is implemented as \( \overline{\overline{A} \cdot \overline{B} \cdot \overline{C}} \).
For the final output \( y \), you combine all product terms using a NAND gate.
Here's the outline of the NAND-NAND implementation:
- Compute each product term using NAND gates.
- Use additional NAND gates to combine the outputs of these product terms to get the final output \( y \).
### 2. Boolean Equation: \( y = PQ + P'QR + PQR \)
#### Convert to Standard SOP Form
The given equation \( y = PQ + P'QR + PQR \) is already in SOP form:
- **Term 1**: \( PQ \)
- **Term 2**: \( P'QR \)
- **Term 3**: \( PQR \)
#### Implement using NAND-NAND Logic
To implement this equation using NAND gates:
1. **Write each term as a NAND expression:**
- **Term 1**: \( PQ \)
- Implement as \( \overline{\overline{P \cdot Q}} \).
- **Term 2**: \( P'QR \)
- Compute \( P' \) using NAND gates: \( P' = P \text{ NAND } P \).
- Then, \( P' \cdot Q \cdot R \) is implemented as \( \overline{\overline{P' \cdot Q \cdot R}} \).
- **Term 3**: \( PQR \)
- Implement as \( \overline{\overline{P \cdot Q \cdot R}} \).
2. **Combine the terms using NAND gates:**
- Implement the OR operation of the terms using NAND gates.
- The final OR operation is \( \overline{\overline{PQ} \cdot \overline{P'QR} \cdot \overline{PQR}} \).
Here's the outline for the NAND-NAND implementation:
- Compute each product term using NAND gates.
- Use additional NAND gates to combine the outputs of these product terms to get the final output \( y \).
### Summary
For both equations, the process involves converting the given Boolean expressions into NAND gate implementations by first expressing the terms as NAND operations and then combining them using NAND gates to get the final output.
### 1. Boolean Equation: \( y = A'B'C' + ABC'D' + A'C'D' \)
#### Convert to Standard SOP Form
The given equation \( y = A'B'C' + ABC'D' + A'C'D' \) is already in the standard SOP form since it is a sum (OR) of products (ANDs) of literals. Each term in the sum is a product of literals, and this is the definition of SOP form.
- **Term 1**: \( A'B'C' \)
- **Term 2**: \( ABC'D' \)
- **Term 3**: \( A'C'D' \)
#### Implement using NAND-NAND Logic
To implement this equation using NAND gates, follow these steps:
1. **Write each term as a NAND expression:**
For NAND gate implementation, first express each AND and OR operation using NAND gates:
- **NAND Gate**: \( A \cdot B \) is implemented as \( \overline{\overline{A \cdot B}} \), where \( \overline{A \cdot B} \) is the NAND operation.
- **OR Gate**: \( A + B \) is implemented as \( \overline{\overline{A} \cdot \overline{B}} \).
2. **Implement the product terms:**
- **Term 1**: \( A'B'C' \)
- First, find \( A' = \overline{A} \), \( B' = \overline{B} \), and \( C' = \overline{C} \) using NAND gates: \( A' = A \text{ NAND } A \), and similarly for \( B \) and \( C \).
- Then, \( A' \cdot B' \cdot C' \) is implemented as \( \overline{\overline{A' \cdot B'} \cdot \overline{C'}} \).
- **Term 2**: \( ABC'D' \)
- Find \( C' = \overline{C} \) and \( D' = \overline{D} \).
- Then, \( A \cdot B \cdot C' \cdot D' \) is implemented as \( \overline{\overline{A \cdot B \cdot C' \cdot D'}} \).
- **Term 3**: \( A'C'D' \)
- Use \( A' = \overline{A} \), \( C' = \overline{C} \), and \( D' = \overline{D} \).
- Then, \( A' \cdot C' \cdot D' \) is implemented as \( \overline{\overline{A' \cdot C' \cdot D'}} \).
3. **Combine the terms using NAND gates:**
To sum the products, you need to OR the terms together. This can be done using NAND gates:
- Implement OR operation by combining the results of the NAND gates for the product terms.
- The OR of three terms \( A, B, \) and \( C \) is implemented as \( \overline{\overline{A} \cdot \overline{B} \cdot \overline{C}} \).
For the final output \( y \), you combine all product terms using a NAND gate.
Here's the outline of the NAND-NAND implementation:
- Compute each product term using NAND gates.
- Use additional NAND gates to combine the outputs of these product terms to get the final output \( y \).
### 2. Boolean Equation: \( y = PQ + P'QR + PQR \)
#### Convert to Standard SOP Form
The given equation \( y = PQ + P'QR + PQR \) is already in SOP form:
- **Term 1**: \( PQ \)
- **Term 2**: \( P'QR \)
- **Term 3**: \( PQR \)
#### Implement using NAND-NAND Logic
To implement this equation using NAND gates:
1. **Write each term as a NAND expression:**
- **Term 1**: \( PQ \)
- Implement as \( \overline{\overline{P \cdot Q}} \).
- **Term 2**: \( P'QR \)
- Compute \( P' \) using NAND gates: \( P' = P \text{ NAND } P \).
- Then, \( P' \cdot Q \cdot R \) is implemented as \( \overline{\overline{P' \cdot Q \cdot R}} \).
- **Term 3**: \( PQR \)
- Implement as \( \overline{\overline{P \cdot Q \cdot R}} \).
2. **Combine the terms using NAND gates:**
- Implement the OR operation of the terms using NAND gates.
- The final OR operation is \( \overline{\overline{PQ} \cdot \overline{P'QR} \cdot \overline{PQR}} \).
Here's the outline for the NAND-NAND implementation:
- Compute each product term using NAND gates.
- Use additional NAND gates to combine the outputs of these product terms to get the final output \( y \).
### Summary
For both equations, the process involves converting the given Boolean expressions into NAND gate implementations by first expressing the terms as NAND operations and then combining them using NAND gates to get the final output.
To convert Boolean equations to Standard Sum-of-Products (SOP) form and implement them using NAND-NAND logic, follow these steps:
### (I) Y = A'B'C' + ABC'D' + A'C'D'
#### 1. Standard SOP Form
The given Boolean equation is already in SOP form, where each term is a product (AND operation) of literals, and these products are summed (OR operation). To ensure it's in standard SOP form, each product term should include all variables in the expression.
Here, the equation is:
\[ Y = A'B'C' + ABC'D' + A'C'D' \]
We need to include all variables (A, B, C, D) in each term, so we use the following steps:
1. **Expand the Terms:**
- \( A'B'C' \) can be expanded to \( A'B'C'D + A'B'C'D' \) (adding \( D \) and \( D' \)).
- \( ABC'D' \) is already expanded.
- \( A'C'D' \) can be expanded to \( A'C'D' \) (adding \( B \) and \( B' \)).
Combining these, the standard SOP form is:
\[ Y = A'B'C'D + A'B'C'D' + ABC'D' + A'B'C'D' + A'B'C'D + A'B'C'D' \]
After combining like terms:
\[ Y = A'B'C'D + A'B'C'D' + ABC'D' + A'C'D' \]
#### 2. Implementing using NAND-NAND Logic
To implement this using NAND gates, follow these steps:
1. **Convert SOP to NAND-NAND form:**
- First, express the SOP form in terms of NAND gates. For any Boolean equation \( F = \overline{\overline{F}} \), where \( \overline{F} \) is the NAND representation of \( F \).
2. **Construct the NAND equivalent:**
Each AND gate can be implemented using NAND gates by De Morgan's theorem. Similarly, OR gates can be implemented using NAND gates.
For example:
- \( A \cdot B = \overline{ \overline{A \cdot B} } \) can be implemented as \( \overline{\overline{A} \cdot \overline{B}} \).
Convert each product term and OR operation into NAND gates:
- \( A'B'C' = \overline{\overline{A'B'C'}} = \overline{(\overline{A'} + \overline{B'} + \overline{C'})} \)
- \( ABC'D' = \overline{\overline{A \cdot B \cdot C' \cdot D'}} = \overline{\overline{A} + \overline{B} + C + D} \)
- \( A'C'D' = \overline{\overline{A' \cdot C' \cdot D'}} = \overline{A + \overline{C} + \overline{D}} \)
Finally, OR the product terms using NAND gates:
- To OR terms \( T_1, T_2, T_3, \ldots \), use \( \overline{\overline{T_1} \cdot \overline{T_2} \cdot \ldots} \).
### (II) Y = PQ + P'QR + PQR
#### 1. Standard SOP Form
This Boolean equation is also in SOP form. Each term is already a product of literals.
The given equation is:
\[ Y = PQ + P'QR + PQR \]
This is already in standard SOP form.
#### 2. Implementing using NAND-NAND Logic
To implement this using NAND gates:
1. **Convert SOP to NAND-NAND form:**
- Apply De Morgan's theorem to convert AND and OR operations to NAND operations.
Each term:
- \( PQ = \overline{\overline{PQ}} = \overline{\overline{P} + \overline{Q}} \)
- \( P'QR = \overline{\overline{P'QR}} = \overline{P + \overline{Q} + \overline{R}} \)
- \( PQR = \overline{\overline{PQR}} = \overline{\overline{P} + \overline{Q} + \overline{R}} \)
To OR the terms, use:
- \( Y = \overline{\overline{PQ} \cdot \overline{P'QR} \cdot \overline{PQR}} \)
Construct the NAND gates to achieve this.
### Summary
- For \( Y = A'B'C' + ABC'D' + A'C'D' \), convert each product and sum to NAND gates.
- For \( Y = PQ + P'QR + PQR \), express each term in NAND and then combine them.
Both tasks involve expressing the given Boolean functions using NAND gates by applying De Morgan's laws to convert AND and OR operations into NAND operations.
### (I) Y = A'B'C' + ABC'D' + A'C'D'
#### 1. Standard SOP Form
The given Boolean equation is already in SOP form, where each term is a product (AND operation) of literals, and these products are summed (OR operation). To ensure it's in standard SOP form, each product term should include all variables in the expression.
Here, the equation is:
\[ Y = A'B'C' + ABC'D' + A'C'D' \]
We need to include all variables (A, B, C, D) in each term, so we use the following steps:
1. **Expand the Terms:**
- \( A'B'C' \) can be expanded to \( A'B'C'D + A'B'C'D' \) (adding \( D \) and \( D' \)).
- \( ABC'D' \) is already expanded.
- \( A'C'D' \) can be expanded to \( A'C'D' \) (adding \( B \) and \( B' \)).
Combining these, the standard SOP form is:
\[ Y = A'B'C'D + A'B'C'D' + ABC'D' + A'B'C'D' + A'B'C'D + A'B'C'D' \]
After combining like terms:
\[ Y = A'B'C'D + A'B'C'D' + ABC'D' + A'C'D' \]
#### 2. Implementing using NAND-NAND Logic
To implement this using NAND gates, follow these steps:
1. **Convert SOP to NAND-NAND form:**
- First, express the SOP form in terms of NAND gates. For any Boolean equation \( F = \overline{\overline{F}} \), where \( \overline{F} \) is the NAND representation of \( F \).
2. **Construct the NAND equivalent:**
Each AND gate can be implemented using NAND gates by De Morgan's theorem. Similarly, OR gates can be implemented using NAND gates.
For example:
- \( A \cdot B = \overline{ \overline{A \cdot B} } \) can be implemented as \( \overline{\overline{A} \cdot \overline{B}} \).
Convert each product term and OR operation into NAND gates:
- \( A'B'C' = \overline{\overline{A'B'C'}} = \overline{(\overline{A'} + \overline{B'} + \overline{C'})} \)
- \( ABC'D' = \overline{\overline{A \cdot B \cdot C' \cdot D'}} = \overline{\overline{A} + \overline{B} + C + D} \)
- \( A'C'D' = \overline{\overline{A' \cdot C' \cdot D'}} = \overline{A + \overline{C} + \overline{D}} \)
Finally, OR the product terms using NAND gates:
- To OR terms \( T_1, T_2, T_3, \ldots \), use \( \overline{\overline{T_1} \cdot \overline{T_2} \cdot \ldots} \).
### (II) Y = PQ + P'QR + PQR
#### 1. Standard SOP Form
This Boolean equation is also in SOP form. Each term is already a product of literals.
The given equation is:
\[ Y = PQ + P'QR + PQR \]
This is already in standard SOP form.
#### 2. Implementing using NAND-NAND Logic
To implement this using NAND gates:
1. **Convert SOP to NAND-NAND form:**
- Apply De Morgan's theorem to convert AND and OR operations to NAND operations.
Each term:
- \( PQ = \overline{\overline{PQ}} = \overline{\overline{P} + \overline{Q}} \)
- \( P'QR = \overline{\overline{P'QR}} = \overline{P + \overline{Q} + \overline{R}} \)
- \( PQR = \overline{\overline{PQR}} = \overline{\overline{P} + \overline{Q} + \overline{R}} \)
To OR the terms, use:
- \( Y = \overline{\overline{PQ} \cdot \overline{P'QR} \cdot \overline{PQR}} \)
Construct the NAND gates to achieve this.
### Summary
- For \( Y = A'B'C' + ABC'D' + A'C'D' \), convert each product and sum to NAND gates.
- For \( Y = PQ + P'QR + PQR \), express each term in NAND and then combine them.
Both tasks involve expressing the given Boolean functions using NAND gates by applying De Morgan's laws to convert AND and OR operations into NAND operations.