### Digital Logic Design: Function of 3 Variables and Implementations#### Problem StatementThe function of 3 variables is specified as a sum of minterms: \[ F(A, B, C) = \Sigma(0, 1, 3, 4, 5) \]#### Tasks1. **Specify the Function F as a Product of Maxterms.**2. **Specify the Complement to the Function as a Sum of Minterms (For F').**3. **Obtain the Schematics for the Following Implementations of Function F:** - **3c1)** Using a 3-to-8 decoder with active-high outputs and a single 3-input logic gate. - **3c2)** Using a 3-to-8 decoder with active-low outputs and a single 3-input logic gate. The available gates are OR3, NOR3, AND3, and NAND3.#### Instructions for Schematics***Note:*** Denote the decoder inputs as:- **A2** for the Most Significant Bit (MSB)- **A1** for the middle bit- **A0** for the Least Significant Bit (LSB) Assign the variables A, B, and C to the decoder inputs. The decoder outputs should be denoted with decimal digits from 0 to 7, which represent minterm or maxterm numbers according to the decoder output type.**Available Logic Gates:**- OR Gate with 3 inputs (OR3)- NOR Gate with 3 inputs (NOR3)- AND Gate with 3 inputs (AND3)- NAND Gate with 3 inputs (NAND3)#### Detailed Explanation:**1. Product of Maxterms for F:** To express F as a product of maxterms, find the complementary minterms (i.e., minterms that are not included in the sum of minterms for F). \[ F'(A, B, C) = \Sigma(2, 6, 7) \] Then, complement F' to find F as a product of maxterms.**2. Sum of Minterms for F' (Complement of F):** The complement of F, denoted as F’, is determined by the minterms not included in the definition of F. \[ F'(A, B, C) = \Sigma(2, 6, 7

To specify the function F(A, B, C) as a product of max terms, we need to find the max terms…

3. The function of 3 variables is specified as a sum of minterms: F (A, B, C) = (0, 1, 3, 4, 5). 3a) Specify the function F as a product of Maxterms. 3b) Specify the complement to the function as a sum of minterms (for F'). 3c) Obtain the schematics for the following implementations of function F: 3c1) with a 3-to-8 decoder with active-high outputs and a single 3-input logic gate.

3. The function of 3 variables is specified as a sum of minterms: F (A, B, C) = (0, 1, 3, 4, 5). 3a) Specify the function F as a product of Maxterms. 3b) Specify the complement to the function as a sum of minterms (for F'). 3c) Obtain the schematics for the following implementations of function F: 3c1) with a 3-to-8 decoder with active-high outputs and a single 3-input logic gate.

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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Question

### Digital Logic Design: Function of 3 Variables and Implementations

#### Problem Statement
The function of 3 variables is specified as a sum of minterms:
\[ F(A, B, C) = \Sigma(0, 1, 3, 4, 5) \]

#### Tasks
1. **Specify the Function F as a Product of Maxterms.**
2. **Specify the Complement to the Function as a Sum of Minterms (For F').**
3. **Obtain the Schematics for the Following Implementations of Function F:**
- **3c1)** Using a 3-to-8 decoder with active-high outputs and a single 3-input logic gate.
- **3c2)** Using a 3-to-8 decoder with active-low outputs and a single 3-input logic gate. The available gates are OR3, NOR3, AND3, and NAND3.

#### Instructions for Schematics
***Note:*** Denote the decoder inputs as:
- **A2** for the Most Significant Bit (MSB)
- **A1** for the middle bit
- **A0** for the Least Significant Bit (LSB)

Assign the variables A, B, and C to the decoder inputs. The decoder outputs should be denoted with decimal digits from 0 to 7, which represent minterm or maxterm numbers according to the decoder output type.

**Available Logic Gates:**
- OR Gate with 3 inputs (OR3)
- NOR Gate with 3 inputs (NOR3)
- AND Gate with 3 inputs (AND3)
- NAND Gate with 3 inputs (NAND3)

#### Detailed Explanation:

**1. Product of Maxterms for F:**
To express F as a product of maxterms, find the complementary minterms (i.e., minterms that are not included in the sum of minterms for F).

\[ F'(A, B, C) = \Sigma(2, 6, 7) \]

Then, complement F' to find F as a product of maxterms.

**2. Sum of Minterms for F' (Complement of F):**
The complement of F, denoted as F’, is determined by the minterms not included in the definition of F.

\[ F'(A, B, C) = \Sigma(2, 6, 7

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