Number System & Boolean Algebra

8/8/2019 Number System & Boolean Algebra

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NUMBER SYSTEM &

BOOLEAN ALGEBRA

Presented by

S Mohanty


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Number System

Why Number System is required?

What are the basic types of Number System?

- Non-Positional

- Positional

What are the types of Positional Number system?

- Decimal

- Binary- Octal

- Hexadecimal


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Non-Positional

Additive approach.

Symbols are used which represents same

value regardless their position in the numberand they are added to find out the value of a

number.


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Positional

Digits are used to represent different values,

depending upon the position they occupy in

the number.

The value of each digit can be determined as:

- the digit itself

- the position of the digit in the no.

- the base/radix of the no. system.


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Decimal Number System

Base=10

At most 10 digits can be used to represent

any decimal no. i.e. 0 to 9. Each position of digit in a decimal no.

represents a power of the base (10).


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Binary Number System

Base=2

At most 2 digits can be used to represent any

binary no. i.e. 0 or 1. Each position of digit in a binary no.

represents a power of the base (2).


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Octal Number System

Base=8

At most 8 digits can be used to represent any

octal no. i.e. 0 to 7. Each position of digit in a octal no. represents

a power of the base (8).

3 bits are used to represent any octal no. in

the computer memory.


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Hexadecimal Number System

Base=16

At most 16 digits can be used to represent

any hexadecimal no. i.e. 0 to 9 of the decimal

no. and the remaining six digits are denoted

by the letters A, B, C, D, E, F.

Each position of digit in a hexadecimal no.

represents a power of the base (16). 4 bits are used to represent any hexadecimal

no. in the computer memory.


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Conversion from one number system to

another Any base no. to decimal no.

Decimal no. to any base no. (Division-

Remainder Method)

Base other than decimal no. to base other

than decimal no.

Binary to Octal & Vice-versa

Binary to Hexadecimal & Vice-versa


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Binary Arithmetic

Addition

1. 0+0=0

2. 0+1=13. 1+0=1

4. 1+1=0 with a carry 1 to the next higher

column.


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Exercise

101 10011 100111

+ 10 +1001 +11011


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Subtraction

1. 0-0=0

2. 1-1=0

3. 1-0=14. 0-1=1 with a borrow 1 from the next higher

column.


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Exercise

10101 1011100

- 01110 - 0111000


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Additive Method of Subtraction

(Complementary Subtraction) Complement of a no.=

[ (Base)n 1] Given no.

where, n-> no. of digits present in a given

no.


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Exercise

Find the complement of following nos.

1. (37)10

2. (6)83. (10101)2


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Steps in Complementary Subtraction

Find the complement of subtrahend.

Add the complement to minuend.

If there is a carry of 1, then add it to theobtained result or if there is no carry,

re-complement the sum add a ve sign to the

result.


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Exercise:

(92)10 (56)10

(18)10 (35)10

(1011100)2 (0111000)2 (010010)2 (100011)2

(10101)2 (01110)2


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Multiplication

0x0=0

1x0=0

0x1=0 1x1=1


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Exercise

1010-> Multiplicand 1111

X 1001-> Multiplier x 111


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Division

0/1=0

1/1=1


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Exercise

100001 / 110= 0101 with remainder 11


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Additive Method ofDivision

(Complementary Subtraction Method) Divisor subtracted from Dividend until the

result of subtraction becomes


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Exercise

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Boolean Algebra

Deals with Binary no. system.

Useful in designing logic circuits which are

used by the processors of computer system

to perform arithmetic operations.

Developed by English Mathematician

Gorge Boole during mid of 18th century.


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Fundamental Concepts ofBoolean

Algebra Use of Binary digits

Logical Addition operation

Logical Multiplication Complementation

Operator Precedence


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Operator Precedence

The Algebraic Exp. should be scanned fromLeft to Right.

Expressions enclosed within parentheses are

evaluated first. All complement operations are performed

next.

All AND or . operations are performed next.

All OR or + operations are performed in thelast.


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Examples ofOperator Precedence

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

= A+(B.C)

IfA

=1, B=0, C=0 then first exp produces 0and second exp produces 1.

Justify which exp is correct.


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Postulates ofBoolean Algebra

A=0 iffA!=1

A=1 iffA!=0

A+0=A

A.1=A

A+B=B+A (Commutative Law over Addition) A.B=B.A (Commutative Law over Multiplication)

A+(B+C)= (A+B)+C (Associative Law over Addition)

A.(B.C)=(A.B).C (Associative Law over Multiplication)

A.(B+C)=(A.B)+(A.C) (Distributive Law over Multiplication)

A+(B.C)=(A+B).(A+C) (Distributive Law over Addition)

A+A=1

A.A=0


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Principle ofDuality

Any theorem in Boolean Algebra has its dual

results by interchanging + with . and 0 with

1.

1+1=1 0.0=0

1+0=0+1=1 0.1=1.0=0

0+0=0 1.1=1


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Theorems ofBoolean Algebra

1. Idempotent Law(a) A+A=A

(b) A.A=A

2. (a) A+1=1

(b) A.0=0

3. Absorption Law(a) A+A.B=A

(b) A.(A+B)=A

4. Involution Law

(A)=A

5. (a) A.(A+B)=A.B

(b) A+A.B=A+B

6. De Morgans Law

(a) (A+B)=A.B

(b) (A.B)= A+B


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BasicBoolean Identities

Sr. No Identities Dual Identities1 A+0=A A.1=A

2 A+1=1 A.0=0

3 A+A=A A.A=A

4 A+A=1 A.A=0

5 (A)=A -

6 A+B=B+A A.B=B.A

7 (A+B)+C=A+(B+C) (A.B).C=A.(B.C)

8 A.(B+C)=A.B+A.C A+(B.C)=(A+B).(A+C)9 A+(A.B)=A A.(A+B)=A

10 A+(A.B)=A+B A.(A+B)=A.B

11 (A+B)=A.B (A.B)=A+B


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Boolean Functions

A Boolean Function is an expression which is

formed with binary variables, two binary

operators i.e. OR and AND, a unary operator

i.e. NOT, parentheses and equal sign.

Example: W=X+(Y.Z)


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Examples

1. x+x.y

2. x.(x+y)

3. x.y.z+x.y.z+x.y4. x.y+x.z+y.z

5. (x+y).(x+z).(y+z)


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Example:

F= x.y.z+x.y.z

F1= x.(y.z+y.z)


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Canonical Forms forBoolean Algebra

Minterms(mj)- AND terms

Maxterms(Mj)- OR terms

Sum-of Products (SOP):

(a) Construct the TT for the given Boolean Function.

(b) Form a minterm for each combination of the

variables which produces 1 in the function.

(c) The desired exp. is sum (OR) of all the minterms

obtained in step-2.


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Exercise:

1. Express the Boolean Function into SOP:

F= A+B.C

2. Express the Boolean Function into POS:f=x.y + x.z


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Logic Gates

All operations within a computer system

carried out by means of combination of

signals passing through built-in circuits,

known as Logic Gate.

Logic Gates are electronic ccts, operate on

one or more inputs and produce standard

outputs. Logic Gates are building blocks of all the

circuits in a computer.


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AND Gate

Logical Multiplication operation.

Generates an o/p 1, iff all inputs are 1.

Truth Table:

A B C=A.B

0 0 0

1 1 0

1 0 01 1 1

Logic Diagram:


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ORGate

Logical Addition operation.

Generates an o/p 1, iff any input is 1.

Truth Table:

A B C=A+B

0 0 0

1 1 1

1 0 11 1 1

Logic Diagram:


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NOT Gate

Complementation operation (Unary operation).

Generates an o/p which is the reverse of the

input.

Truth Table:

A A

0 1

1 0

Logic Diagram:


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NAND Gate

Complemented AND Gate.

Generates an o/p 1, iff all and any one input is 0 andgenerates an o/p 0, iff all inputs are 1.

Truth Table:A B C=(A.B)=A+B

0 0 1

1 1 1

1 0 11 1 0

Logic Diagram:


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NORGate

Complemented OR Gate.

Generates an o/p 1, iff all inputs are 0 andgenerates an o/p 0, iff any input is 1.

Truth Table:A B C=(A+B)=A.B

0 0 1

1 1 0

1 0 01 1 0

Logic Diagram:


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Exclusive-ORGate

Denoted by

Generates an o/p 1, iff both inputs are different andgenerates an o/p 0, iff both inputs are same.

Truth Table:A B C=(A B)=A.B+A.B

0 0 0

1 1 1

1 0 11 1 0

Logic Diagram:


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Exclusive-NORGate

Denoted by

Generates an o/p 1, iff both inputs are same andgenerates an o/p 0, iff both inputs are different.

Truth Table:A B C=(A B)=(A B)=(A.B+A.B)=A.B+ A.B

0 0 1

1 1 0

1 0 01 1 1

Logic Diagram:


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NAND as Universal Gate

The following logical operations can be

performed with the implementation of NAND

Gates:

NOT Gate

AND Gate

OR Gate

Ex-OR Gate Ex-NOR Gate


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NORas Universal Gate

The following logical operations can be

performed with the implementation of NOR

Gates:

NOT Gate

AND Gate

OR Gate

Ex-OR Gate Ex-NOR Gate


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Assignment-2

Exercise:

Draw the logic circuit of EX-OR operation by

using NOT, AND, OR gates.

Draw the logic circuit NOT, AND, OR, EX-

OR, EX-NOR operations by using NAND

gates only.

Draw the logic circuit NOT, AND, OR, EX-

OR, EX-NOR operations by using NOR gatesonly.


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Assignments

Write the procedures/steps to convert anybase no. system to decimal no. system alongwith examples.

Write the procedures/steps to convert anydecimal no. system to any base no. systemalong with examples.

Write the procedures/steps to convert any

base other than decimal no. system to anybase other than decimal no. system alongwith examples.


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Contd

Prove all the enlisted theorems of Boolean

Algebra in the previous slide by using

Boolean Postulates or Perfect Induction

Method.

Prepare a presentation upon this topic for my

next class. Note that each presentation topic

should be different among syndicates.


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Documents

Number System & Boolean Algebra