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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
35/5
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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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