Chapter 2

Boolean Algebra and Logic Gates

Boolean algebra is an algebraic structure defined by a set of elements, B,

together with two binary operators, + and. B is defined as a set with only

two elements, 0 and 1.

BASIC THEOREMS AND PROPERTIES OF BOOLEAN ALGEBRA:

Proof the following

1

2

By using truth table:

BOOLEAN FUNCTIONS

A Boolean function described by an algebraic expression consists of

binary variables, the constants 0 and 1, and the logic operation symbols.

A schematic of an implementation of the function:

3

Solution:

Simplify Boolean Function:

4

Complement of a Function

The complement of a function F is F_ and is obtained from an

interchange of 0’s for 1’s and 1’s for 0’s in the value of F. The

complement of a function may be derived algebraically through

DeMorgan’s theorems.

DeMorgan’s theorems:

states that the complement of a function is obtained by interchanging

AND and OR operators and complementing each literal.

Example:

Solution:

5

CANONICAL AND STANDARD FORMS:

Minterms and Maxterms:

1. Each of AND terms is called a minterm, or a standard product.

2. Each of OR terms is called maxterms, or standard sums.

A Boolean function can be expressed algebraically from a given truth

table by forming a minterm for each combination of the variables that

produces a 1 in the function and then taking the OR of all those terms.

Example: Find Minterm & Maxterm from the table below

Solution:

Since each one of these minterms results in f1 = 1

6

The complement of f1 or each one of these Maxterm results in f1 = 0

Canonical form:

Boolean functions expressed as a sum of minterms or product of

maxterms.

Sum of Minterms:

any Boolean function can be expressed as a sum of minterms. The

minterms whose sum defines the Boolean function are those which give

the 1’s of the function in a truth table.

If the function is not in this form, it can be made so by first expanding

the expression into a sum of AND terms. Each term is then inspected to

see if it contains all the variables. If it misses one or more variables, it is

ANDed with an expression such as

7

8

From the truth table, we can then read the five minterms of the function

to be 1, 4, 5, 6, and 7.

Product of Maxterms:

To express a Boolean function as a product of maxterms, it must first be

brought into a form of OR terms. This may be done by using the

distributive law,

Conversion between Canonical Forms:

The complement of a function expressed as the sum of minterms equals

the sum of minterms missing from the original function. This is because

the original function is expressed by those minterms which make the

function equal to 1, whereas its complement is a 1 for those minterms

for which the function is a 0.

To convert from one canonical form to another, interchange the symbols

∑ and П and list those numbers missing from the original form

Example: consider the complement of the following function:

9

Solution:

Example: Consider the Boolean expression

Solution:

10

Chapter 3

Gate-Level Minimization

THE MAP METHOD

The map method presented for use minimizing Boolean functions. The

map method is also known as the Karnaugh map or K-map .

Definition: A K-map is a diagram made up of squares, with each square

representing one minterm of the function that is to be minimized.

Since any Boolean function can be expressed as a sum of minterms.

Type of K- map:

1. Two-Variable K-Map:

Example1: by using K- map simplify the

F(X,Y)=

2.

Three-

Variable K-Map:

11

3. Four-Variable K-Map:

Example simplify the bolean function by suing K- map

F (W, X, Y, Z)= ∑(0,1, 4, 5, 12, 13, 8, 9)

Solution

12