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Chapter 12: Problem 20

Find the boolean expression represented by each Karnaugh map.

Short Answer

Expert verified
Given a Karnaugh Map, find the boolean expression by following these steps: 1. Identify the prime implicants and essential prime implicants by grouping the largest power-of-two configurations of 1's in the K-map. 2. Write the boolean expression for each prime implicant. 3. Simplify the boolean function using essential prime implicants and any additional prime implicants necessary to cover all 1's. 4. Combine the boolean expressions using the OR operation to obtain the final boolean expression. Provide the given K-map to find the specific boolean expression.

Step by step solution

01

Identify the given Karnaugh map (K-map)

In this case, we are not provided with the K-map, but we can go through a generic process of determining boolean expressions from K-maps. When provided with a K-map, please follow the following steps.
02

Identify prime implicants and essential prime implicants

Inspect the K-map for groups of 1's, which can be combined in the largest possible power-of-two configurations (1, 2, 4, 8, etc.). These groups can be rectangles or squares, and the K-map can wrap around the edges. Ensure to choose the largest possible groups, as larger groups will result in more simplified expressions. These groups are called prime implicants. To identify an essential prime implicant, look for any 1鈥檚 on the K-map that are included in only one prime implicant. The prime implicants that cover these unique 1's are essential prime implicants.
03

Write the boolean expression for each prime implicant

For each prime implicant, identify the input variables that remain unchanged within the group. Write down the input variable if it stays at value 1 and the negation of the input variable if it stays at value 0. For example, if we have a group of 1's in a 2-variable K-map where the rows are A and columns are B, and the group covers the cells where A=1 and B=0,1, then the boolean expression for this prime implicant is just A because B changes, but A remains the same.
04

Simplify the boolean function using essential prime implicants

First, include all essential prime implicants in your final boolean expression. If there are still uncovered 1's in the K-map (that is, not covered by essential prime implicants), select the remaining prime implicants in such a manner that they cover all other 1's with as few additional prime implicants as possible.
05

Present the final boolean expression

Combine the boolean expressions for all essential prime implicants and any additional prime implicants chosen in step 4 using the OR operation. This resulting expression is the boolean function represented by the given Karnaugh map. Please provide the given Karnaugh Map so that we can help you find the boolean expression in accordance with the above steps.

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Most popular questions from this chapter

The set \(D_{70}=\\{1,2,5,7,10,14,35,70\\}\) of positive factors of 70 is a boolean algebra under the operations \(\oplus, \odot,\) and ' defined by \(x \oplus y=\operatorname{lcm}\\{x, y\\}\) \(x \odot y=\operatorname{gcd}\\{x, y\\},\) and \(x^{\prime}=70 / x .\) Compute each. $$7^{\prime} \oplus 2^{\prime}$$

Simplify each boolean expression using the laws of boolean algebra. $$w x y z+w^{\prime} x y^{\prime} z^{\prime}+w x y z^{\prime}+w^{\prime} x y^{\prime} z$$

Find the dual of each boolean property. $$(x+y) z=x z+y z$$

Define the operations \(+, \cdot,\) and \(^{\prime}\) on \(B=\\{0,1\\}\) as follows: \(x+y=\) \(\max \\{x, y\\}, x \cdot y=\min \\{x, y\\}, 0^{\prime}=1,\) and \(1^{\prime}=0 .\) Is \(\left\langle B,+, \cdot,,^{\prime}, 0,1\right\rangle \mathrm{a}\) boolean algebra?

Give all minterms three boolean variables \(x, y,\) and \(z\) can generate.
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