by: miss farah adibah adnan imk. chapter outline: part iv 1.4 boolean algebras 1.4.1 introduction...

BY: MISS FARAH ADIBAH ADNANBY: MISS FARAH ADIBAH ADNAN

IMKIMK

CHAPTER OUTLINE: PART IV

1.4 BOOLEAN ALGEBRAS

1.4.1 INTRODUCTION1.4.2 BOOLEAN OPERATIONS 1.4.3 BOOLEAN EXPRESSIONS1.4.4 BOOLEAN FUNCTIONS1.4.5 DUALITY1.4.6 IDENTITIES OF BOOLEAN ALGEBRA

1.4.5 LOGIC NETWORKS

1.4.1 INTRODUCTION1.4.1 INTRODUCTION

Boolean algebra – used in design circuits in computers and other electronic devices.

The operation of a circuits is defined by a Boolean expression – specify an output for each set of inputs.

It deals with values of 0 and 1 / SET {0,1}.Called – bits, binary digitsThere are 3 operations : AND, OR, and NOT.

1.4.2 BOOLEAN OPERATIONS 1.4.2 BOOLEAN OPERATIONS 1.4.2.1 ANDAlso known as Boolean Product.Denoted – dot (.)Has the following values:

1.1=1, 1.0=0, 0.1=0, 0.0=0

1.4.2.2 ORAlso known as Boolean ProductDenoted – sum (+)Has the following values:

1+1=1, 1+0=1, 0+1=1, 0+0=0

1.4.2 BOOLEAN OPERATIONS 1.4.2 BOOLEAN OPERATIONS 1.4.2.3 NOTAlso known as complement.Denoted – bar ( ), negation ( ), (‘)It interchanges 0 and 1.

x

X Y XY X+Y

X’

0 0 0 0 1

1 0 0 1 0

0 1 0 1 1

1 1 1 1 0

Truth Table for AND, OR, NOT operations:

Example 1.1Find the value of

Answer :

1.0 (0 1)

0 1 0 0

0

1.4.3 BOOLEAN 1.4.3 BOOLEAN EXPRESSIONSEXPRESSIONSThe Boolean Expressions in the variables

are defined as follows:

If are Boolean expressions, then

are Boolean expression.

Example 1.2:

1)

2)

1 2, ,..., nx x x

1 20,1, , ,..., nx x x and X Y , ,X Y

, ( )X Y XY

( , , ) ( ' )P x y z x y z ( , , ) ( ' )Q x y z x y z

1.4.4 BOOLEAN 1.4.4 BOOLEAN FUNCTIONSFUNCTIONSEach Boolean expression represents a Boolean

function. The values of this function are obtained by substituting

0 and 1 for the variables in the expression. Tables that listing the values of function f for all

elements of are often called the truth table for f .

Example 1.3:

Find the values of the Boolean function represented by

.

( , , )F x y z xy z

nB

1.4.5 DUALITY1.4.5 DUALITYThe dual of any statement in a Boolean

algebra – is the statement obtained by:1)Interchanging the operations (+) and (.)2)Interchanging their identity elements 0

and 1 in the original statement

Example 1.4:

Find the dual of .bba )1*()*0(

1.4.6 IDENTITIES OF BOOLEAN 1.4.6 IDENTITIES OF BOOLEAN ALGEBRAALGEBRA

Boolean algebra satisfies many of the same laws as ordinary algebra - addition and multiplication.

The following laws are common to both kinds of algebra:

1.4.7 LOGIC NETWORKS1.4.7 LOGIC NETWORKSA computer or other electric device is made up of a

number of circuits.Each circuit can be designed using the rules of

Boolean algebra. The basic element of circuits – gates.There are 3 basic types of gates: Inverter, OR gate, and

AND gate.

1.4.7 LOGIC NETWORKS1.4.7 LOGIC NETWORKS

1) Inverter/NOT: Accept one Boolean variable as input, and produces

the complement of this value as output.

A A’

1 0

0 1

1.4.7 LOGIC NETWORKS1.4.7 LOGIC NETWORKS

2) OR gate: The inputs to this gate are the values of two or more

Boolean variables, while the output is the sum of their values.

A B A+B

1 1 1

1 0 1

0 1 1

0 0 0

1.4.7 LOGIC NETWORKS1.4.7 LOGIC NETWORKS

3) AND gate: The inputs to this gate are the values of two or more

Boolean variables, while the output is the Boolean product of their values.

A B AB

1 1 1

1 0 0

0 1 0

0 0 0

1.4.7 LOGIC NETWORKS1.4.7 LOGIC NETWORKS

COMBINATION OF GATES

1) NAND Gate This is a NOT-AND gate which is equal to

an AND gate followed by a NOT gate.

*The outputs of all NAND gates are high if any of the inputs are

low.

A B

1 1 0

1 0 1

0 1 1

0 0 1

AB

1.4.7 LOGIC NETWORKS1.4.7 LOGIC NETWORKS

COMBINATION OF GATES

1) NOR Gate This is a NOT-OR gate which is equal to an

OR gate followed by a NOT gate.

*The outputs of all NOR gates are low if any of the inputs are

high.

A B

1 1 0

1 0 0

0 1 0

0 0 1

A B

Consider a function of three variables x, y, and z.Since each variable may be complemented or

uncomplemented, there are different combinations.

When combinations are combined with AND, they are called Minterms.

When Combinations are combined with OR, they are called Maxterms.

Minterm Canonical Form – Standard Products.Maxterm Canonical Form - Standard Sums.

1.4.8 MINTERM CANONICAL 1.4.8 MINTERM CANONICAL FORMFORM

32 8

1.4.8 MINTERM CANONICAL 1.4.8 MINTERM CANONICAL FORMFORM

For n Variables there are 2^n Minterms/Maxterms

Determine the Set of Minterms for which a function is 1-valued. These are called “Minterms of the Function”

Combine all Minterms with a + OperationThe sum of minterms that represents the function is

called – the sum of products expansion.

1.4.8 MINTERM CANONICAL 1.4.8 MINTERM CANONICAL FORMFORM

1.4.8 MINTERM CANONICAL 1.4.8 MINTERM CANONICAL FORMFORM

• The product of maxterms that represents the function is called – the product-of-sum expansion.

Example 1.5

Find the sum-of-product expansion for the function

.

1.4.8 MINTERM CANONICAL 1.4.8 MINTERM CANONICAL FORMFORM

, ,F x y z x y z