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Question 5: Minimize the following boolean function K map: F(A, B, C, D) = Em(0, 2, 8, 10, 14) + Ed(5, 15)

Question 5: Minimize the following boolean function K map: F(A, B, C, D) = Em(0, 2, 8, 10, 14) + Ed(5, 15)

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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Minimize the following boolean function K map: F(A, B, C, D) = Σm(0, 2, 8, 10, 14) + Σd(5, 15)
### Question 5: Minimize the following Boolean function K-map **Problem Statement:** Minimize the following Boolean function K-map: \[ F(A, B, C, D) = \sum m(0, 2, 8, 10, 14) + \sum d(5, 15) \] ### Explanation: In this problem, you are provided with a Boolean function \( F \) that is a function of four variables \( A, B, C, \) and \( D \). The function is expressed in terms of minterms \(\sum m\) and don't-care conditions \(\sum d\). #### Key Points: - **Minterms** (\(\sum m\)): These are the specific combinations of variables \( (A, B, C, D) \) for which the function \( F \) is equal to 1. For this problem, the minterms are 0, 2, 8, 10, and 14. - **Don't-care conditions** (\(\sum d\)): These are combinations where the function's value doesn't affect the outcome and can be used to simplify the expression. For this problem, the don't-care conditions are 5 and 15. ### Steps to Minimize: 1. **Construct a Karnaugh Map (K-map)**: - Create a 4-variable K-map (16 cells). - Populate the K-map with the provided minterms and don't-care conditions. 2. **Simplify using K-map**: - Group the 1’s (from minterms) and X’s (don't-care conditions) into the largest possible groups (power of 2: 1, 2, 4, 8...). - Derive the simplified Boolean expression from the groups. ### K-map Layout for 4 variables (ABCD): | | CD\AB | 00 | 01 | 11 | 10 | |-------|-------|----|----|----|----| | **00** | | 0 | 1 | X | 1 | | **01** | | 0 | 1 | 0 | X | | **11** | | X | 1 | 1 | 0 | | **10** | | 1 | X | 0 | 1 | - **0**
Transcribed Image Text:### Question 5: Minimize the following Boolean function K-map **Problem Statement:** Minimize the following Boolean function K-map: \[ F(A, B, C, D) = \sum m(0, 2, 8, 10, 14) + \sum d(5, 15) \] ### Explanation: In this problem, you are provided with a Boolean function \( F \) that is a function of four variables \( A, B, C, \) and \( D \). The function is expressed in terms of minterms \(\sum m\) and don't-care conditions \(\sum d\). #### Key Points: - **Minterms** (\(\sum m\)): These are the specific combinations of variables \( (A, B, C, D) \) for which the function \( F \) is equal to 1. For this problem, the minterms are 0, 2, 8, 10, and 14. - **Don't-care conditions** (\(\sum d\)): These are combinations where the function's value doesn't affect the outcome and can be used to simplify the expression. For this problem, the don't-care conditions are 5 and 15. ### Steps to Minimize: 1. **Construct a Karnaugh Map (K-map)**: - Create a 4-variable K-map (16 cells). - Populate the K-map with the provided minterms and don't-care conditions. 2. **Simplify using K-map**: - Group the 1’s (from minterms) and X’s (don't-care conditions) into the largest possible groups (power of 2: 1, 2, 4, 8...). - Derive the simplified Boolean expression from the groups. ### K-map Layout for 4 variables (ABCD): | | CD\AB | 00 | 01 | 11 | 10 | |-------|-------|----|----|----|----| | **00** | | 0 | 1 | X | 1 | | **01** | | 0 | 1 | 0 | X | | **11** | | X | 1 | 1 | 0 | | **10** | | 1 | X | 0 | 1 | - **0**
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