Unit 01 2

**Unit 1** **Unit 1**

### Introduction Introduction

### Number Systems and Number Systems and

### Conversion

### Conversion

Unit 01 3

**Unit 1** **Unit 1**

zz

**Introduction** **Introduction**

**Number Systems and Conversion** **Number Systems and Conversion**

1.1 Digital Systems and Switching Circuits1.1 Digital Systems and Switching Circuits

1.2 Number Systems and Conversion1.2 Number Systems and Conversion

1.3 Binary Arithmetic1.3 Binary Arithmetic

1.4 Representation of Negative Numbers1.4 Representation of Negative Numbers

9Addition of 29Addition of 2’’s Complement Numberss Complement Numbers 99Addition of 1Addition of 1’’s Complement Numberss Complement Numbers

1.5 Binary Codes1.5 Binary Codes

Unit 01 4

**Digital Systems and Switching Circuits** **Digital Systems and Switching Circuits**

zz

### Digital system Digital system

The physical quantities or signals can The physical quantities or signals can assume only

*assume only discretediscrete* valuesvalues

Greater accuracyGreater accuracy zz

### Analog system Analog system

The physical quantities or signals may vary The physical quantities or signals may vary
*continuously*

*continuously* over a specified rangeover a specified range

Unit 01 5

Unit 01 6

**Digital Systems and Switching Circuits** **Digital Systems and Switching Circuits**

zz Design of digital systemsDesign of digital systems

System designSystem design

99Breaking the overall system into subsystemsBreaking the overall system into subsystems 99Specifying the characteristics of each subsystemSpecifying the characteristics of each subsystem

9 E.g. digital computer : 9E.g. digital computer : memory units, arithmetic unit, I/O *memory units, arithmetic unit, I/O *
*devices, control unit*

*devices, control unit*

Logic designLogic design

99Determining how to interconnect basic logic building blocks to Determining how to interconnect basic logic building blocks to perform a specific function

perform a specific function

9 E.g. arithmetic unit : binary addition: 9E.g. arithmetic unit : binary addition: logic gates, Flip*logic gates, Flip-**-Flops, **Flops, *
*interconnections*

*interconnections*

Circuit designCircuit design

99Specifying the interconnection of specific components such as Specifying the interconnection of specific components such as resistors, diodes, and transistors to form a gate, flip

resistors, diodes, and transistors to form a gate, flip--flop or other flop or other logic building block

logic building block

9 E.g. Flip9E.g. Flip--Flop: Flop: resistors, diodes, transistors*resistors, diodes, transistors*

Unit 01 7

**Digital Systems and Switching Circuits** **Digital Systems and Switching Circuits**

zz Many of subsystems of a digital system take the Many of subsystems of a digital system take the form of a switching network

form of a switching network

Switching NetworksSwitching Networks

99Combinational NetworksCombinational Networks

•• No memoryNo memory

99Sequential NetworksSequential Networks

•• Combinational Circuits + MemoryCombinational Circuits + Memory

Unit 01 8

**Number Systems and Conversion** **Number Systems and Conversion**

zz

### Positional notation Positional notation Base 10:

### Base 10:

### Base 2:

### Base 2:

Unit 01 9

**Number Systems and Conversion** **Number Systems and Conversion**

zz

### Base Base *R* *R* : :

### Any positive integer

### Any positive integer *R* *R* ( ( *R* *R* >1) can be chosen as >1) can be chosen as the radix or base of a number system.

### the radix or base of a number system.

where .
where .^{0} ^{≤} ^{a}^{i}^{≤} ^{R}^{−} ^{1}

Unit 01 10

**Number Systems and Conversion** **Number Systems and Conversion**

### Example:

### Example:

For bases greater than 10, more than 10 symbols are needed For bases greater than 10, more than 10 symbols are needed to represent the digits. In hexadecimal (base 16),

to represent the digits. In hexadecimal (base 16), AA presents presents
1010_{10}_{10}, B, B presents 11presents 11_{10}_{10}, C , C presents 12presents 12_{10}_{10}, D, D presents 13presents 13_{10}_{10}, E, E

presents 14

presents 14_{10}_{10}, F, F presents 15presents 15_{10}_{10}..

Unit 01 11

**Number Systems and Conversion** **Number Systems and Conversion**

zz

### Convert a decimal Convert a decimal *integer* *integer* to base to base *R* *R*

### This process is continued until we finally This process is continued until we finally obtain

### obtain *a* *a*

_{n}*..*

_{n}2 3

3 1

4 4

1 2 3

1 2

2 1

3 3

1 1 2

0 1

1 1

2 2

1 1

remaider

,

remaider

,

remaider

,

*a*
*Q*

*a*
*R*

*a*
*R*

*a*
*R*

*R* *a*
*Q*

*a*
*Q*

*a*
*R*

*a*
*R*

*a*
*R*

*R* *a*
*Q*

*a*
*Q*

*a*
*R*

*a*
*R*

*a*
*R*

*R* *a*
*N*

*n*
*n*

*n*
*n*

*n*
*n*

*n*
*n*

*n*
*n*

*n*
*n*

= +

+ +

+

=

= +

+ +

+

=

= +

+ +

+

=

−

−

−

−

−

−

−

−

−

L L L

Unit 01 12

**Number Systems and Conversion** **Number Systems and Conversion**

zz Example : Convert 53Example : Convert 53_{10}_{10} to binary.to binary.

Unit 01 13

**Number Systems and Conversion** **Number Systems and Conversion**

zz

### Convert a decimal fraction to base Convert a decimal fraction to base *R* *R*

### This process is continued until we have This process is continued until we have obtained a

### obtained a *sufficient* *sufficient* number of digits. number of digits.

Unit 01 14

**Number Systems and Conversion** **Number Systems and Conversion**

zz

### Example: Convert 0.625 Example: Convert 0.625

_{10}

_{10}

### to binary. to binary.

Unit 01 15

**Number Systems and Conversion** **Number Systems and Conversion**

zz Example: Convert 0.7Example: Convert 0.7_{10 }_{10 }to binary.to binary.

Unit 01 16

**Number Systems and Conversion** **Number Systems and Conversion**

zz Example: Convert 231.3Example: Convert 231.3_{4}_{4} to base 7.to base 7.

Unit 01 17

**Number Systems and Conversion** **Number Systems and Conversion**

zz

### Conversion from binary to hexadecimal Conversion from binary to hexadecimal ( and conversely)

### ( and conversely)

One hexadecimal digit corresponds to four One hexadecimal digit corresponds to four
*bibinary digit*nary digi*ts (bit*s (*bit*s)s)

*D*
*D*

6 0

3

) 1101 0110

0000 0011

( )

. 306

( _{16} = ⋅ _{2}

Unit 01 18

**Binary Arithmetic** **Binary Arithmetic**

zz AdditionAddition

Example: Add 13Example: Add 13_{10}_{10} and 11and 11_{10}_{10} in binary.in binary.

Unit 01 19

**Binary Arithmetic** **Binary Arithmetic**

zz SubtractionSubtraction

Examples:Examples:

Unit 01 20

**Binary Arithmetic** **Binary Arithmetic**

zz

### Multiplication Multiplication

Example:Example:

Unit 01 21

**Binary Arithmetic** **Binary Arithmetic**

zz

### Division Division

Example: 145/11=13 Example: 145/11=13 --- 22

Unit 01 22

**Representation of Negative Numbers** **Representation of Negative Numbers**

zz

### 2 2 ’ ’ s Complement Number System s Complement Number System

Positive Number Positive Number *N N *

99 *N **N *is represented by a 0 followed by the is represented by a 0 followed by the
magnitude.

magnitude.

Negative Number Negative Number *––NN*

99*–**–**N **N *is represented by its 2is represented by its 2’’s complement, s complement,
*N**N**. If the word length is n,*. If the word length is n,

*NN**=2*=2^{n}^{n}*--NN*

Unit 01 23

**Representation of Negative Numbers** **Representation of Negative Numbers**

zz 11’’s Complement Number Systems Complement Number System

Positive Number Positive Number *N **N *

99 *N **N *is represented by a 0 followed by the is represented by a 0 followed by the
magnitude

magnitude

Negative Number Negative Number *–**–**N**N*

9–9*–N **N is represented by its 1*is represented by its 1’’s complement . s complement .
If the word length is

If the word length is *n**n*,,

*N*=2**N**=2^{n}* ^{n}*-N-

*N*=(2=(2

^{n}*--11--*

^{n}*N)+1= +1*

*N*)+1= +1

*N*

*N* *N* = ( 2

^{n}### − 1 ) −

*N*

Unit 01 24

**Representation of Negative Numbers** **Representation of Negative Numbers**

zz

### Sign and Magnitude Binary Numbers Sign and Magnitude Binary Numbers

Unit 01 25

**Addition of 2**

**Addition of 2** **’** **’** **s Complement Numbers** **s Complement Numbers**

zz Addition of Addition of *nn--bit signedbit signed* *binary numbersbinary numbers*

Any carry from the sign position is ignored.Any carry from the sign position is ignored.

n=4n=4

Unit 01 26

**Addition of 2**

**Addition of 2** **’** **’** **s Complement Numbers** **s Complement Numbers**

).

( is result the

so

, 2 g subtractin to

equivalent is

carry last away the Throwing

carry) , (

2 ) (

2

) 2

(

n

*

*A*
*B*

*A*
*B*
*A*

*B*

*B*
*A*
*B*

*A*
*B*
*A*

*n*
*n*

*n*

−

>

≥

− +

=

+

−

= +

= +

−

Unit 01 27

**Addition of 2**

**Addition of 2** **’** **’** **s Complement Numbers** **s Complement Numbers**

).

( of tion representa correct

the is which

, ) (

) (

2 yields carry last the Discarding

carry) ,

2 (

2 ) (

2 2

) 2 ( ) 2 (

* 1

*

*

*B*
*A*

*B*
*A*
*B*
*A*
*B*

*A*
*B*

*A*

*B*
*A*

*B*
*A*
*B*
*A*

*n*
*n*
*n*

*n*
*n*

*n*
*n*

+

−

+

= +

−

≤ +

≥ +

− +

=

− +

−

= +

=

−

−

−

Unit 01 28

**Addition of 2**

**Addition of 2** **’** **’** **s Complement Numbers** **s Complement Numbers**

Example: Add

Example: Add --8 and +19 in 28 and +19 in 2’’s complements complement for a word length of n=8.

for a word length of n=8.

Unit 01 29

**Addition of 1**

**Addition of 1** **’** **’** **s Complement Numbers** **s Complement Numbers**

zz Addition of Addition of *nn--bit signedbit signed* *binary numbersbinary numbers*

*Add the **Add the **last carry**last carry* *( **( **end**end**-**-**around carry**around carry**) to the n-**) to the n**-*
*bit sum in the position furthest to the right.*

*bit sum in the position furthest to the right.*

*n* = 4

Unit 01 30

**Addition of 1**

**Addition of 1** **’** **’** **s Complement Numbers** **s Complement Numbers**

).

( is result the

so

, 1 adding and

2 g subtractin to

equivalent is

carry around

- end The

carry) , (

2 1 ) (

2

) 1 2 (

n

*A*
*B*

*A*
*B*
*A*

*B*

*B*
*A*
*B*

*A*
*B*
*A*

*n*
*n*

*n*

−

>

≥

−

− +

=

+

−

−

= +

= +

−

Unit 01 31

**Addition of 1**

**Addition of 1** **’** **’** **s Complement Numbers** **s Complement Numbers**

).

( of tion representa correct

the is which

, ) (

) (

1 2 yields carry last the Discarding

carry) ,

2 (

2 1 )]

( 1 2 [ 2

) 1 2 ( ) 1 2 (

1

*B*
*A*

*B*
*A*
*B*

*A*
*B*
*A*
*B*

*A*

*B*
*A*

*B*
*A*
*B*
*A*

*n*

*n*
*n*

*n*
*n*

*n*
*n*

+

−

+

= +

−

−

<

+

≥

− +

−

− +

=

−

− +

−

−

= +

=

−

−

−

Unit 01 32

**Addition of 1**

**Addition of 1** **’** **’** **s Complement Numbers** **s Complement Numbers**

Example: Add

Example: Add --11 and -11 and -20 in 120 in 1’’s complement for s complement for a word length of n=8.

a word length of n=8.

Unit 01 33

**Binary Codes** **Binary Codes**

zz Weighted codeWeighted code

*w**w*_{3}_{3}*-w**-**w*_{2}_{2}*-w**-**w*_{1}_{1}*-w**-**w*_{0}* _{0}* weighted code weighted code

*a*

*a*

_{3}

_{3}*a*

*a*

_{2}

_{2}*a*

*a*

_{1}

_{1}*a*

*a*

_{0}

_{0}*a*

*a*

_{3}

_{3}*a*

*a*

_{2}

_{2}*a*

*a*

_{1}

_{1}*a*

*a*

_{0}

_{0}*= w*

*= w*

_{3}

_{3}*a*

*a*

_{3}

_{3}*+w*

*+w*

_{2}

_{2}*a*

*a*

_{2}

_{2}*+w*

*+w*

_{1}

_{1}*a*

*a*

_{1}

_{1}*+w*

*+w*

_{0}

_{0}*a*

*a*

_{0}

_{0} Binary-Binary-Coded-Coded-Decimal, BCD; 8Decimal, BCD; 8--44--22--1BCD code1BCD code 0101=0

0101=0^{.}^{.}8+18+1^{.}^{.}4+04+0^{.}^{.}2+12+1^{.}^{.}1=51=5

6-6-3-3-11--1 code1 code 0101=0

0101=0^{.}^{.}6+16+1^{.}^{.}3+03+0^{.}^{.}1+11+1^{.}^{.}1=41=4

zz Excess-Excess-3 code3 code

8-8-4-4-22--1 code + 00111 code + 0011

* The code of i*The code of

*is the 1’is the 1’s complement of code*

**i**

**s complement of code 9**

**9-**

**-i**

**i**z

z 2-2-outout--ofof--5 code5 code

Exactly 2 out of 5 bits are 1Exactly 2 out of 5 bits are 1

Error-Error-checking propertieschecking properties zz Gray codeGray code

The codes for successive decimal digits differ in exactly on bit.The codes for successive decimal digits differ in exactly on bit. zz ASCII codeASCII code

**American ****A**merican S**Standard **tandard C**C**ode forode for**I****I**nformation Information **Interchange**nterchange

Unit 01 34

**Binary Codes**

**Binary Codes**

Unit 01 35

**Binary Codes**

**Binary Codes**

Unit 01 36

*Homework* *Homework*

### z z 1.1 (b); 1.1 (b);

### z z 1.2 (b); 1.2 (b);

### z z 1.4 (c); 1.4 (c);

### z z 1.5 (b); 1.5 (b);

### z z 1.7 (a), (c); 1.7 (a), (c);

### z z 1.9 1.9

### z z 1.16 (b); 1.16 (b);

### z z 1.17 (b); 1.17 (b);

### z z 1.24 1.24

### z z 1.25 (a); 1.25 (a);

**Unit 2** **Unit 2**

**Boolean**

**Boolean** **Algebra** **Algebra**

Unit 02 2

**Unit 2** **Unit 2**

zz

**Boolean Algebra** **Boolean Algebra**

2.1 Introduction2.1 Introduction

2.2 Basic Operations2.2 Basic Operations

2.3 Boolean Expressions and Truth Tables2.3 Boolean Expressions and Truth Tables

2.4 Basic Theorems2.4 Basic Theorems

2.5 Commutative, Associative, and 2.5 Commutative, Associative, and Distributive Laws

Distributive Laws

2.6 Simplification Theorems2.6 Simplification Theorems

2.7 Multiplying Out and Factoring2.7 Multiplying Out and Factoring

2.8 2.8 DemorganDemorgan’’ss LawsLaws

Unit 02 3

**Introduction** **Introduction**

zz **George George BooleBoole** **developed Boolean algebra developed Boolean algebra **
**in 1847 and used it to solve problems in **
**in 1847 and used it to solve problems in **

**mathematical logic.**

**mathematical logic.**

zz **Boolean AlgebraBoolean Algebra**

A Boolean algebraA **Boolean algebra***is an algebra*is an *algebra*
**(B; . , + ,**

**(B; . , + ,’’** **;0, 1);0, 1)** consisting of a set consisting of a set **BB** (which (which
contains at least two elements

contains at least two elements **00** and and **11**) )
together with

together with *threethree* operations,operations,

the the **ANDAND** *(Boolean product*(Boolean *product*) operation ) operation **..** ,,
the the **OROR** *(Boolean sum*(Boolean *sum*) operation ) operation **++**, and, and

the the **NOTNOT** *(complement*(*complement*) operation ) operation **’’** , defined , defined
on the set, such that

on the set, such that

Unit 02 4

**Introduction** **Introduction**

A0. ClosureA0. Closure: For any : For any x,yx,y, and z of B, , and z of B, *x**x*^{.}^{.}*y**y**, **, **x**x***+****+***y**y**, x**, x***’*** ’* are in Bare in B

A1. IdempotentA1. Idempotent: : xx^{.}^{.}*x**x**=x **=x **x+x**x+x**=x**=x*

A2. CommutativeA2. Commutative: : *x**x*^{.}^{.}*y**y**=**=**y**y*^{.}^{.}*x**x* *xy**xy**=**=**yx**yx*

A3. AssociativeA3. Associative: :

xx^{.}*^{.}* ((yy

^{.}*zz)=()=(xx*

^{.}

^{.}*yy) )*

^{.}

^{.}*z=x z=x*

^{.}

^{.}*y y*

^{.}

^{.}*z z xx*

^{.}

**+***((yy*

**+**

**+***zz)=()=(xx*

**+**

**+***yy) )*

**+**

**+***z=x z=x*

**+**

**+***y y*

**+**

**+***zz*

**+** A4. AbsorptiveA4. Absorptive: x : x ^{.}*^{.}* (x (x

**+***y)=x x y)=x x*

**+**

**+***(x(x*

**+**^{.}

^{.}y)=xy)=x

A5. DistributiveA5. Distributive::

x x ^{.}*^{.}* (y (y

**+***z)=(x z)=(x*

**+**

^{.}*y) y)*

^{.}

**+***(x (x*

**+**

^{.}*z)z) x x*

^{.}

**+***(y (y*

**+**

^{.}*z)=(x z)=(x*

^{.}

**+***y) y)*

**+**

^{.}*(x (x*

^{.}

**+***z)z)*

**+** A6. ZeroA6. Zero( null, smallest), ( null, smallest), **0****0** , and One, and One( universal, largest), ( universal, largest), **1****1** , ,
elements are in B

elements are in B

x x ^{.}^{.}**1****1**==**1****1** ^{.}*^{.}* x=x x x=x x

**+**

**+****0**

**0**==

**0**

**0**

**+***x=xx=x*

**+** A7. ComplementA7. Complement

For every x in B, there exists a unique x

For every x in B, there exists a unique x’’ in B such thatin B such that
x x ^{.}*^{.}* xx’’==

**0**

**0**, x , x

**+***xx’’==*

**+****1**

**1**

Unit 02 5

**Introduction** **Introduction**

zz **Example:Example:** TwoTwo--element Boolean Algebraelement Boolean Algebra
(Switching Algebra)

(Switching Algebra)
BB_{2}_{2}=({0,1}; =({0,1}; ^{. }^{. },+, ,+, ’’ ; 0,1); 0,1)

AND OR NOT AND OR NOT

11 00

11

00 00

00

11 00

..

11 11

11

11 00

00

11 00

++

00 11

11 00

’’

Unit 02 6

**Introduction** **Introduction**

zz **Example:Example:** FourFour--element Boolean Algebraelement Boolean Algebra
BB_{4}_{4}=({0,a,b,1} ; =({0,a,b,1} ; ^{. }^{. },+, ,+, ’’ ; 0,1) ; 0,1)

AND OR NOT AND OR NOT

11 bb aa 00 11

bb bb 00 00 bb

aa 00 aa 00 aa

00 00 00 00 00

11 bb aa 00 ..

11 11 11 11 11

11 bb 11 bb bb

11 11 aa aa aa

11 bb aa 00 00

11 bb aa 00 ++

00 11

aa bb

bb aa

11 00

’’

Unit 02 7

**Introduction** **Introduction**

zz An element of a Boolean algebra B is An element of a Boolean algebra B is called a

*called a constant constant on B.*on B.

e.g. 0,a,b,1 in B
e.g. 0,a,b,1 in B_{4}_{4}..

zz A symbol that may represent any one of A symbol that may represent any one of element of B is called a

element of B is called a (Boolean) (Boolean)
*variable*

*variable* on B.on B.

e.g.

e.g. x,y,zx,y,z,,……

Unit 02 8

**Introduction** **Introduction**

zz *A Boolean expression*A *Boolean expression* over an algebra over an algebra
system (B;

system (B; ^{. }^{. },+, ,+, ’’ ; 0,1) is defined as ; 0,1) is defined as
follows:

follows:

1.1. Any element of B ( constantAny element of B ( constant) is a Boolean expression.) is a Boolean expression.

2.2. Any variableAny variable name is a Boolean expression.name is a Boolean expression.

3.3. If eIf e_{1}_{1} and eand e_{2 }_{2 }are Boolean expression, then eare Boolean expression, then e_{1}_{1}’, e’, e_{2}_{2}’, e’, e_{1}_{1}+e+e_{2}_{2}, ,
ee_{1}_{1}.e.e_{2}_{2} are Boolean expressions.are Boolean expressions.

4.4. Any expression that can be constructed by a finite Any expression that can be constructed by a finite number of applications of the above rules, and

number of applications of the above rules, and onlyonly such a expression is a Boolean expression.

such a expression is a Boolean expression.

Unit 02 9

**Introduction** **Introduction**

zz A A *function f(x*function *f(x*_{1}_{1}*,x,x*_{2}_{2}*,,……,,xx*_{n}_{n}*)),,*
*from *

*from BB*^{n}^{n}*to B is called a Boolean functionto B is called a Boolean function* *if it if it *
*can be specified by a Boolean expression of *
*can be specified by a Boolean expression of *
*n variables x*

*n variables x*_{1}_{1}*,x,x*_{2}_{2}*,,……,,xx*_{n}_{n}*..*
*f(a,b,c*

*f(a,b,c)=)=abab’’c+ac+a’’b+bb+b’’cc’’*

zz Each appearance of Each appearance of *a variable or its a variable or its *
*complement*

*complement* in an expression is referred to in an expression is referred to
as a

*as a literalliteral*..
*f(a,b,c*

*f(a,b,c)=)=abab’’c+ac+a’’b+bb+b’’c’c’*
has 3 variables,

has 3 variables, *a,ba,b,,* and and *cc*, 7 , 7 literals(literals(*aa, b, b’’* *, c, , c, *
*aa’’, b, b, b, b’’, c, c’’* ).).

*B* *B*

*f* :

^{n}### →

Unit 02 10

**Basic Operations** **Basic Operations**

zz

### The basic operations of Boolean algebra are The basic operations of Boolean algebra are AND, OR, and NOT (complement, or

### AND, OR, and NOT (complement, or inverse)

### inverse) . .

NOT (Complement)NOT (Complement)

InverterInverter 1

0′ = 1′ = 0

1

0

and

0

1 = ′ = =

′ = *if* *X* *X* *if* *X*

*X*

Unit 02 11

**Basic Operations** **Basic Operations**

AND OperationAND Operation

### 9Omit the symbol 9 Omit the symbol “ “

^{.}

^{.}

### ” ” , A , A

^{. }

^{. }

### B=AB B=AB

AND Gate AND Gate

Unit 02 12

**Basic Operations** **Basic Operations**

OR operationOR operation

OR Gate OR Gate

Unit 02 13

**Boolean Expressions and Truth Tables** **Boolean Expressions and Truth Tables**

zz

### Order in which the operations are perform Order in which the operations are perform Parentheses

### Parentheses Æ Æ Complentation Complentation Æ Æ AND AND Æ Æ OR OR

zz

### Circuits for expressions Circuits for expressions

*C*
*B*

*A* ′ +

*BE*
*D*

*C*

*A*( + )]′ +
[

Unit 02 14

**Truth Table** **Truth Table**

zz If an expression has If an expression has *n**n* variables, the number of different variables, the number of different
combinations of values of the variables is

combinations of values of the variables is 2*2*^{n}* ^{n}*. Therefore, a truth . Therefore, a truth
table for n

table for n--variable expression will have variable expression will have *2**2*^{n}* ^{n}* rows.rows.

zz There are functions of There are functions of 2^{(}^{2}^{n}^{)} *n**n* variablesvariables. .

Unit 02 15

**Truth Table** **Truth Table**

zz *AB*′ + *C* = (*A* + *C*)(*B*′ + *C*)

Unit 02 16

**Basic Theorems** **Basic Theorems**

zz

Unit 02 17

**Basic Theorems** **Basic Theorems**

zz

Unit 02 18

**Basic Theorems** **Basic Theorems**

zz

### Switching Circuits Switching Circuits

= =

Unit 02 19

**Commutative, Associative, and ** **Commutative, Associative, and **

**Distributive Laws** **Distributive Laws**

zz

### Commutative Laws Commutative Laws

zz

### Associative Laws Associative Laws

zz

### Distributive Laws Distributive Laws

Unit 02 20

**Multiple**

**Multiple** **-** **-** **Input Gates** **Input Gates**

zz Associative Laws for AND Associative Laws for AND andand OR OR operations.

operations.

(AB)C=A(BC)=ABC (AB)C=A(BC)=ABC

(A+B)+C=A+(B+C)=A+B+C (A+B)+C=A+(B+C)=A+B+C

Unit 02 21

**Proof of Boolean Theorems** **Proof of Boolean Theorems**

zz By Truth TableBy Truth Table

Example: X+YZ=(X+Y)(X+Z)Example: X+YZ=(X+Y)(X+Z)

11 11

11 11

11 11

11 11

11 11

11 11

00 00

11 11

11 11

11 11

00 11

00 11

11 11

11 11

00 00

00 11

11 11

11 11

11 11

11 00

00 00

11 00

00 00

11 00

00 11

00 00

00 11

00 00

00 00

00 00

00 00

00 00

(X+Y)(X+Z) (X+Y)(X+Z) X+ZX+Z

X+YX+Y X+YZX+YZ

YZYZ ZZ

YY XX

Unit 02 22

**Proof of Boolean Theorems** **Proof of Boolean Theorems**

zz

### By basic laws: By basic laws:

**Example:Example:** X+YZ=(X+Y)(X+Z)X+YZ=(X+Y)(X+Z)

Unit 02 23

**Simplification Theorems** **Simplification Theorems**

zz

### Useful Simplification Theorems Useful Simplification Theorems

zz

### Proof Proof

Unit 02 24

**Simplification Examples** **Simplification Examples**

zz **Example: SimplifyExample: Simplify**
Sol:

Sol:

zz Example: Simplify Example: Simplify

Sol:

Sol:

] ) (

][

[ + ′ + + + ′ + + ′

= *A* *B**C* *D* *EF* *A* *B**C* *D* *EF*
*Z*

*C*
*B*
*A*

*X*

*Z* = = + ′

) (

) )(

( + ′ + ′ ′ + + ′

= *AB* *C* *B* *D* *C* *E* *AB* *C*

*Z*

) (

) (

) )(

(

then

, )

( Let

+ ′

′+ + ′

= ′ +

=

′ +

=

+ ′

′ + + ′

+ ′

=

′ = + ′

= ′ + ′

*C*
*AB*
*E*

*C*
*D*
*B*
*Y*
*X*
*Y*
*X*
*Y*

*C*
*AB*
*E*

*C*
*D*
*B*
*C*
*AB*
*Z*

*X*
*E*

*C*
*D*
*B*
*Y*
*C*

*AB*

Unit 02 25

**Simplification Examples** **Simplification Examples**

zz **Example:Example:** Find the output Find the output *YY* of the following of the following
circuit and design a simpler circuit having the
circuit and design a simpler circuit having the

same output.

same output.

Sol:

Sol:

The resulting circuit contains only

The resulting circuit contains only one OR**one OR** gate.gate.

*A*
*B*

*A*
*B*

*B*
*B*
*A*
*A*

*B)B*
*B*

*(A*
* *

*A*
*B))B*

*(AB*
*B*

*(A*
*Y*

+

= +

′ +

= +

′+

=

+ +

′ +

=

Unit 02 26

**Multiplying Out and Factoring** **Multiplying Out and Factoring**

zz SumSum--ofof--products (SOP)products (SOP)

An expression is said to be in

*An expression is said to be in sum**sum**-**-**of**of**-**-**products**products* form when all form when all
products are the

products are the products of only single variables.products of only single variables.

zz ProductProduct--ofof--sums (POS)sums (POS)

An expression is said to be in

*An expression is said to be in product**product-**-of**of-**-sums**sums* form when all form when all
sums are the

sums are the sums of only single variablessums of only single variables..

form.

SOP in (

form, SOP

in

,

*not*
*is*
*EF*
*B)CD*

*A*

*are*
*E*
*D*
*C*
*B*
*A*
*E*

*C*
*A*
*E*
*D*
*C*
*B*
*A*

+ +

+ ′

′+

′ +

′ +

′+

form.

POS in

form, POS

in )

( ,

) )(

)(

(

*EF is not *
*D)*

*B)(C*
*(A*

*are*
*E*

*D*
*C*
*B*
*A*
*E*

*C*
*A*
*E*
*D*

*C*
*B*
*A*

+ +

+

′+

′ + ′

+ ′

′+

′ + +

Unit 02 27

**Multiplying Out and Factoring** **Multiplying Out and Factoring**

zz

*Using the ordinary distributive law * Using the *ordinary distributive law * *X(Y+Z)=XY+XZ*

*X(Y+Z)=XY+XZ*

*to multiply out* to *multiply out* an expression to obtain a an expression to obtain a sum sum - - of of - - products form. products form.

zz

*Using the second distributive law* Using the *second distributive law* *X+YZ=(X+Y)(X+Z)*

*X+YZ=(X+Y)(X+Z)*

*to factor* to *factor* an expression to obtain a an expression to obtain a product

### product - - of of - - sums form. sums form.

Unit 02 28

**Multiplying Out and Factoring** **Multiplying Out and Factoring**

zz

### Example: Multiply out (A+BC)(A+D+E). Example: Multiply out (A+BC)(A+D+E).

### Sol Sol - - 1: 1:

**Multiply out the original expression completely**

**Multiply out the original expression completely**and then eliminating redundant terms:

and then eliminating redundant terms:

(A+BC)(A+D+E)=A+AD+AE+ABC+BCD+BCE (A+BC)(A+D+E)=A+AD+AE+ABC+BCD+BCE

=A(1+D+E+BC)+BCD+BCE

=A(1+D+E+BC)+BCD+BCE

=A+BCD+BCE

=A+BCD+BCE

### Sol Sol - - 2: 2:

**Use (X+Y)(X+Z)=X+YZ**

**Use (X+Y)(X+Z)=X+YZ**::

(A+BC)(A+D+E)=A+BC(D+E)=A+BCD+BCE (A+BC)(A+D+E)=A+BC(D+E)=A+BCD+BCE