Solutions For Digital Design Chapter 3 Problem 16P

Procedural steps:

1. The pair of eight squares denoted by the blue and green rectangle can be expressed as and respectively.

2. Now combine all the expression of each pair to yield the expression in the sum-of-products form. In this problem, the combined expression is .

Therefore the simplified Boolean function is .

To implement the simplified Boolean function with a two-level NAND gate circuits, first need to convert the function into the expression containing NAND operations.

The Boolean function can be rewritten as,

Implementation of the simplified Boolean function obtained in equation (1) with two-level NAND gate circuit is shown in Figure 2.

(b)

Consider the following Boolean function:

The expression can be expressed in sum of minterms as follows.

The Karnaugh map of the Boolean function is shown in Figure 3.

1. Express each pair of adjacent squares by their respective prime implicants. The pair of four vertical squares denoted by the blue rectangle can be expressed as . The pair of four squares at the bottom denoted by the green rectangle can be expressed as and the pair of four squares denoted by the brown rectangles can be expressed as .

2. Now combine all the expression of each pair to yield the expression in the sum-of-products form. In this problem, the combined expression is .

Therefore the simplified Boolean function is .

To implement the simplified Boolean function with a two-level NAND gate circuits, we first need to convert the function into the expression containing NAND operations.

The Boolean function can be rewritten as,

……. (2)

Implementation of the simplified Boolean function obtained in equation (2) with two-level NAND gate circuit is shown in Figure 4.

Desired circuit has implemented.

(c)

Consider the following Boolean function:

To simplify the Boolean function, we complement the function twice.

Taking complement of the given Boolean function, we get,

…… (3)

Again take the complement of the complemented function obtained in equation (3) to get the original function.

Simplify further.

Therefore, the simplified Boolean function is .

To implement the simplified Boolean function with a two-level NAND gate circuits, we first need to convert the function into the expression containing NAND operations.

The Boolean function can be rewritten as,

……. (4)

Implementation of the simplified Boolean function obtained in equation (4) with two-level NAND gate circuit is shown in Figure 5.

Consider the following Boolean function:

Apply distributive postulate of Boolean algebra.

…… (5)

Therefore, the simplified Boolean function is.

To implement the simplified Boolean function with a two-level NAND gate circuits, we first need to convert the function into the expression containing NAND operations.

The Boolean function can be rewritten as,

……. (6)

Implementation of the simplified Boolean function obtained in equation (6) with two-level NAND gate circuit is shown in Figure 6.

Desired circuit has implemented.