Problem

Simplify the following Boolean functions, using three-variable maps: (a) \(F(x, y, z)=\Sig...

Simplify the following Boolean functions, using three-variable maps:

(a) $F(x, y, z)=\Sigma(0,1,5,7)$

(b) $F(x, y, z)=\Sigma(1,2,3,6,7)$

(d) $F(x, y, z)=\Sigma(1,2,3,5,6,7)$

(e) $F(x, y, z)=\Sigma(0,2,4,6)$

(f) $F(x, y, z)=\Sigma(3,4,5,6,7)$

Step-by-Step Solution

Solution 1

Step #1 of 10

(a)

Consider the expression:

The three-variable map of the given Boolean function is shown in Table 1.

Table 1 Three variable map for

$E:\Chegg\Assignments Solution files\Assignment 7\Batch 1\Ch 3\Excel docs & pics\3-2P(a).png$

Step #2 of 10

Steps:

1. The first step is to mark the squares of minterms 000, 001, 101 and 111 by 1’s.

2. The next step is to find the possible adjacent squares. The upper two squares are adjacent surrounded by blue rectangle. Another pair of two adjacent squares is marked by green rectangle.

3. The third step is to express each pair of adjacent squares by the common expression. The pair denoted by blue rectangle can be expressed as . Similarly another pair marked by green rectangle can be expressed as .

4. The next step is to combine all the expression of each pairs to yield in the sum-of-products form. In this problem, the combined expression is .

5. Further simplify the combined expression if needed. In this case, there is no need of doing such thing.

Therefore the simplified Boolean function is .

Step #3 of 10

(b)

Consider the expression:

The three-variable map of the given Boolean function is shown in Table 2.

Table 2: Three variable map for

$E:\Chegg\Assignments Solution files\Assignment 7\Batch 1\Ch 3\Excel docs & pics\3-2P(b).png$

Steps:

1. Mark the squares of minterms 001, 001, 011, 110 and 111 by 1’s.

2. Find the possible adjacent squares. The four right corner squares are adjacent surrounded by blue rectangle. Another pair of two adjacent squares is marked by green rectangle.

3. Express each pair of adjacent squares by the common expression. The pair denoted by blue rectangle can be expressed as . Similarly another pair marked by green rectangle can be expressed as .

4. Combine all the expression of each pairs to yield in the sum-of-products form. In this problem, the combined expression is .

5. Further simplify the combined expression if needed. In this case, there is no need of doing such thing.

Therefore the simplified Boolean function is .

Step #4 of 10

(c)

Consider the expression:

The three-variable map of the given Boolean function is shown in Table 3.

Table 3: Three variable map for

$E:\Chegg\Assignments Solution files\Assignment 7\Batch 1\Ch 3\Excel docs & pics\3-2P(c).png$

Steps:

1. Mark the squares of minterms 010, 011, 100 and 101 by 1’s.

2. Find the possible adjacent squares. The upper two right squares are adjacent surrounded by blue rectangle. Another pair of adjacent squares is marked by green rectangle.

3. Express each pair of adjacent squares by the common expression. The pair denoted by blue rectangle can be expressed as . Similarly another pair marked by green rectangle can be expressed as .

4. Combine all the expression of each pairs to yield in the sum-of-products form. In this problem, the combined expression is .

5. Further simplify the combined expression if needed. In this case, there is no need of doing such thing.

Therefore the simplified Boolean function is .

Step #5 of 10

(d)

Consider the expression:

The three-variable map of the given Boolean function is shown in Table 4.

Table 4: Three variable map for

$E:\Chegg\Assignments Solution files\Assignment 7\Batch 1\Ch 3\Excel docs & pics\3-2P(d).png$

Step #6 of 10

Steps:

1. Mark the squares of minterms 001, 010, 011, 101, 110 and 111 by 1’s.

2. Find the possible adjacent squares. The middle four squares are adjacent surrounded by blue rectangle. Another pair of four adjacent squares is marked by green rectangle.

3. Express each pair of adjacent squares by the common expression. The pair denoted by blue rectangle can be expressed as . Similarly another pair denoted by green rectangle can be expressed as .

4. Combine all the expression of each pairs to yield in the sum-of-products form. In this problem, the combined expression is .

5. Further simplify the combined expression if needed. In this case, there is no need of doing such thing.

Therefore the simplified Boolean function is .

Step #7 of 10

(e)

Consider the expression:

The three-variable map of the given Boolean function is shown in Table 5.

Table 5: Three variable map for

$E:\Chegg\Assignments Solution files\Assignment 7\Batch 1\Ch 3\Excel docs & pics\3-2P(e).png$

Step #8 of 10

Steps:

1. Mark the squares of minterms 000, 010, 100 and 110 by 1’s.

2. Find the possible adjacent squares. There is only one four adjacent squares at each corner marked by blue rectangle.

3. Express the pair of adjacent squares by the common expression. The pair denoted by blue rectangle can be expressed as .

4. Combine all the expression of each pairs to yield in the sum-of-products form. In this problem, since there is only one expression, the combined expression is .

5. Further simplify the combined expression if needed. In this case, there is no need of doing such thing.

Therefore the simplified Boolean function is .

Step #9 of 10

(f)

Consider the expression:

The three-variable map of the given Boolean function is shown in Table 6.

Table 6 Three variable map for

$E:\Chegg\Assignments Solution files\Assignment 7\Batch 1\Ch 3\Excel docs & pics\3-2P(f).png$

Step #10 of 10

Steps:

1. Mark the squares of minterms 011, 100, 101, 101, 110 and 111 by 1’s.

2. Find the possible adjacent squares. The lower four squares are adjacent surrounded by blue rectangle. Another pair of two adjacent squares is marked by green rectangle.

3. Express each pair of adjacent squares by the common expression. The pair denoted by blue rectangle can be expressed as . Similarly another pair denoted by green rectangle can be expressed as.