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1) Simplify the following Boolean functions, using three variable K-maps.

1) Simplify the following Boolean functions, using three variable K-maps.

Database System Concepts
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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Make K-map as the instructions
### Simplification of Boolean Functions using Three Variable K-maps #### 1) Simplify the following Boolean functions, using three variable K-maps. ##### a) F(x, y, z) = Σ(0, 1, 2, 3, 5) To simplify the given Boolean function using a K-map: 1. Construct a 3-variable K-map with the appropriate cells labeled for \(xyz\). 2. Place 1s in the cells corresponding to the min terms given in the function's summation notation. 3. Group the 1s into the largest possible power of two groups (1, 2, 4, 8, etc.). 4. Write down the simplified expression for each group. **Steps:** - First, draw the K-map for three variables: ``` yz | 00 | 01 | 11 | 10 ------------------------- x | ------------------------- 0 x | 1 | 1 | 1 | 1 | ------------------------- 1 x | 1 | 0 | 0 | 1 | ------------------------- ``` - From the K-map groups: (Combine the cells with 1s in the power of two) - Group 1 (Covering the top row entirely): The simplified form would be \( \overline{x} \) - Group 2 (Combining cells (100) and (110)): The simplified form would be \(xy\) Hence, \(F(x, y, z) = \overline{x} + xy \) ##### b) F(x, y, z) = x'y + yz' + y'z' To simplify using a K-map: 1. Construct a 3-variable K-map for \(xyz\). 2. Populate the K-map with 1s for each min term corresponding to the given expression. 3. Group the 1s into the largest possible power of two groups. 4. Write down the simplified expression for each group. **Steps:** - Draw the K-map for three variables: ``` yz | 00 | 01 | 11 | 10 ------------------------- x | ------------------------- 0 x | 1 | 1
Transcribed Image Text:### Simplification of Boolean Functions using Three Variable K-maps #### 1) Simplify the following Boolean functions, using three variable K-maps. ##### a) F(x, y, z) = Σ(0, 1, 2, 3, 5) To simplify the given Boolean function using a K-map: 1. Construct a 3-variable K-map with the appropriate cells labeled for \(xyz\). 2. Place 1s in the cells corresponding to the min terms given in the function's summation notation. 3. Group the 1s into the largest possible power of two groups (1, 2, 4, 8, etc.). 4. Write down the simplified expression for each group. **Steps:** - First, draw the K-map for three variables: ``` yz | 00 | 01 | 11 | 10 ------------------------- x | ------------------------- 0 x | 1 | 1 | 1 | 1 | ------------------------- 1 x | 1 | 0 | 0 | 1 | ------------------------- ``` - From the K-map groups: (Combine the cells with 1s in the power of two) - Group 1 (Covering the top row entirely): The simplified form would be \( \overline{x} \) - Group 2 (Combining cells (100) and (110)): The simplified form would be \(xy\) Hence, \(F(x, y, z) = \overline{x} + xy \) ##### b) F(x, y, z) = x'y + yz' + y'z' To simplify using a K-map: 1. Construct a 3-variable K-map for \(xyz\). 2. Populate the K-map with 1s for each min term corresponding to the given expression. 3. Group the 1s into the largest possible power of two groups. 4. Write down the simplified expression for each group. **Steps:** - Draw the K-map for three variables: ``` yz | 00 | 01 | 11 | 10 ------------------------- x | ------------------------- 0 x | 1 | 1
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