**Using Karnaugh Maps for 7-Segment Display Outputs**To simplify logic expressions for a 7-segment display, we can use Karnaugh maps (K-maps). Below, we focus on outputs 'b' and 'e'.### Simplifying SOP Expression for Output 'b'Output 'b' corresponds to the second column under the output values in the truth table. For each row where 'b' is 1, plot the corresponding values of $x_3, x_2, x_1,$ and $x_0$ on the K-map. Merge the groups to find the simplest Sum of Products (SOP) expression.### Simplifying POS Expression for Output 'e'Output 'e' corresponds to the sixth column. For simplifying the Product of Sums (POS), plot 0s on the K-map. Group the 0s to find the simplest POS expression.### Truth Table ExplanationThe truth table included in the image shows:| $x_3$ | $x_2$ | $x_1$ | $x_0$ | a | b | c | d | e | f | g ||---------|---------|---------|---------|---|---|---|---|---|---|---|| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 || 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 || 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 || 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 || 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 || 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 || 0 |

Answered: Use a Karnaugh map to simplify the SOP…

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

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**Using Karnaugh Maps for 7-Segment Display Outputs**

To simplify logic expressions for a 7-segment display, we can use Karnaugh maps (K-maps). Below, we focus on outputs 'b' and 'e'.

### Simplifying SOP Expression for Output 'b'
Output 'b' corresponds to the second column under the output values in the truth table. For each row where 'b' is 1, plot the corresponding values of $x_3, x_2, x_1,$ and $x_0$ on the K-map. Merge the groups to find the simplest Sum of Products (SOP) expression.

### Simplifying POS Expression for Output 'e'
Output 'e' corresponds to the sixth column. For simplifying the Product of Sums (POS), plot 0s on the K-map. Group the 0s to find the simplest POS expression.

### Truth Table Explanation

The truth table included in the image shows:

| $x_3$ | $x_2$ | $x_1$ | $x_0$ | a | b | c | d | e | f | g |
|---------|---------|---------|---------|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 |
| 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 |
| 0 |

Expert Solution

Step 1

Given data,

Here $x_{3}, x_{2}, x_{1}, a n d x_{0}$ are the input variable.

And $a, b, c, d, e, f, a n d g$ are the output variable

Here the truth table is given, and the aim of the given question is to find the specified output in SOP (sum of product) or POS (Product of sum) form using the K-map (Karnaugh map).

K-map

In many digital circuits and practical problems, finding output expressions with minimum variables are necessary.

The Boolean expressions of 3, 4 variables can minimize very easily using K-map without using any Boolean algebra theorems.

K-map can take two forms Sum of Product (SOP) and Product of Sum (POS) according to the need of the problem.

K-map is a table-like representation, but it gives more information than TRUTH TABLE.

The grid of the K-map is filled with 0’s and 1’s then solves it by making groups.

Steps to solve expression using K-map-

Select K-map according to the number of variables.
Identify minterms or maxterms as given in the problem.
For SOP put 1’s in blocks of K-map respective to the minterms (0’s elsewhere).
For POS put 0’s in blocks of K-map respective to the maxterms(1’s elsewhere).
Make rectangular groups containing total terms in the power of two like 2,4,8 ..(except 1) and try to cover as many elements as you can in one group.
From the groups made in step 5 find the product terms and sum them up for SOP form.

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