boolean algebra saras m. srivastava pgt (computer science) kendriya vidyalaya, iisc bangalore –...

BOOLEAN

ALGEBRA

Saras M. SrivastavaPGT (Computer Science)Kendriya Vidyalaya, IISc

Bangalore – 560 012

What is logic ?

Any logical things have only two state either

true or false.

Example :

Is 12 – 10 = 2 Yes

Is India post populated country in the world?

No

Saras M. Srivastava PGT CS, KV IISc, Bangalore 12

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BINARY VALUED QUANTITIES

Binary Decision : THE DECISION which results into either YES

/TRUE OR NO/FALSE

Tautology : If the result of any logical statement or

expression is always TRUE or 1

Fallacy : If the result is always FALSE or 0


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BINARY VALUED QUANTITIES

Binary Decision : THE DECISION which results into either YES

/TRUE OR NO/FALSE

Tautology : If the result of any logical statement or

expression is always TRUE or 1

Fallacy : If the result is always FALSE or 0


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LOGICAL OPERATIONS

Logic operations

NOT (Complement)- Changes TRUE to FALSE, 1 to 0, high to low.

AND — “A AND B” is true if A and B are individually true (A•B, AB, A.B)

OR — “A OR B” is true if either A or B is true (or both are true) (A+B, A.B).


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Evaluation of Boolean Expression using truth table

Prepare truth table X’Y’ + X’Y XY’(Z+YZ’) + Z’ Prove using truth table (X + Y)’ = X’Y’ X + XY = X


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LOGIC GATES

What is logic Gate ?Gate is an electronic circuit which operates on

one or more signals to produce an output

signal. Basically there are three type of Logic

Gate.


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TRUTH TABLES Truth table Represents all the possible

values of logical variables / statements along with all the possible results of the given combination of values.


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Inverter (NOT Gate)

This Gate operates on single variable and operation performed by NOT operator called complementation. Thus A means complement of A’.

A A’

0 1

1 0

Input Output

A A’

Symbol

Truth Table


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AND GATE:

If both inputs signals are 1 (i.e. high) then the output will also be 1 (i.e. high).

Otherwise, the output willbe 0 (i.e. low).

A B A.B

0 0 0

0 1 0

1 0 0

1 1 1

A

BA.B

Truth Table

Two input AND Gate

The AND Gate have two or more than two input signals and produce an out put signal.


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OR Gate:

If either of the two input signals are 1 (high), or both of them are 1 (high), the output will be 1 (high).

A B A+B

0 0 0

0 1 1

1 0 1

1 1 1

AB

A+B

The OR Gate have two or more than two input signals and produce an out put signal.


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Universal Gates

NAND and NOR Gate are known as Universal Gate

because any Boolean function can be constructed

using only NAND or only NOR gates.

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The NAND Gate:

The NAND gate is a combination of an AND gate followed by an inverter (not Gate). It is compliment of AND Gate.

A B Y

0 0 1

0 1 1

1 0 1

1 1 0A

B Y

The NAND (Not AND) Gate has two or more input signal but only one output signal. If All the inputs are 1 (i.e. high), then the output l is 0 (i.e. low)


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15NAND Gates into Other Gates:-

AY

A

B

Y

YAB

NOT Gate

AND Gate

OR Gate

We can create other gates with the help of NAND Gate.


NOR Gate :The NOR (Not OR) Gate has two or more input signal but only one output signal. If All the inputs are 0 (i.e. low), then the output signal is 1 (i.e. high)

A B Y

0 0 1

0 1 0

1 0 0

1 1 0

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AB

Y

The NOR gate is a combination of an OR gate followed by an inverter (not Gate). It is compliment of OR Gate.

Truth Table for NOR Gate

XOR Gate (Exclusive OR Gate):- We use XOR Gate for parity check. XOR Gate produces output 1 for only those inputs combinations that have odd number of 1’s.

# Odd number of 1’s produce output 1. Two input XOR Gate

A B Y

0 0 0

0 1 1

1 0 1

1 1 0

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AB

Y=AB’+A’B

# A B= AB’+A’B

Truth Table for XOR Gate :

+

XNOR Gate (Exclusive NOR Gate):- We use XNOR Gate for parity check. XNOR Gate produces output 1 for only those inputs combinations that have even number of 1’s.

# Even number of 1’s and 0’s produce output 1.

Truth Table for Two Input XNOR Gate

A B Y

0 0 1

0 1 0

1 0 0

1 1 1

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A B=AB+A’B’

Laws &

Theorems of

Boolean

Algebra

Boolean Algebra A Boolean algebra is defined as a closed

algebraic system containing a set K (two or more) elements and two operators, and +.

Operators + and are similar to + and x. Useful for identifying and minimizing circuit

functionality Identity elements

a + 0 = a a 1 = a

0 is the identity element for the + operation.

1 is the identity element for the operation.


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Duality

The principle of duality says that if an expression is valid in Boolean algebra, the dual of that expression is also valid.

To form the dual of an expression, replace all + operators with operators, all operators with + operators, all ones with zeros, and all zeros with ones.

Form the dual of the expressiona + (b c) = (a + b) (a + c)

Following the replacement rules…a (b + c) = a b + a c


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Commutativity and Associativity

The Commutative Property:For every a and b in K, a + b = b + a a b = b a

The Associative Property:For every a, b, and c in K, a + (b + c) = (a + b) + c a (b c) = (a b) c


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Distributivity and Complements

The Distributive Property:For every a, b, and c in K, a ( b + c ) = ( a b ) + ( a c ) a + ( b c ) = ( a + b ) ( a + c )

The Existence of the Complement:For every a in K there exists a unique element called a’ (complement of a) such that, a + a’ = 1 a a’ = 0


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Involution

This theorem states:a’’ = a

Taking the double inverse of a value will give the initial value.


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Absorption

This theorem states:a + ab = a a(a+b) = a

To prove the first half of this theorem:a + ab = a . 1 + ab

= a (1 + b)= a (b + 1)= a (1)

a + ab = a


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DeMorgan’s Theorem

A key theorem in simplifying Boolean algebra expression is DeMorgan’s Theorem. It states: (a + b)’ = a’b’and (ab)’ = a’ + b’


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Truth Table to Boolean Expression

Converting a truth table to an expression Each row with output of 1 becomes a product term Sum product terms (SOP) together.

x00001111

y00110011

z01010101

G00010011

xyz + xyz’ + x’yz

Any Boolean Expression can be represented in sum of products form!


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Equivalent Circuit Representations Number of 1’s in truth table output column equals AND terms for Sum-of-Products (SOP)

x y z

x00001111

y00110011

z01010101

G00010011

G = xyz + xyz’ + x’yz

G

x xx

xx

x

xx

x


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Minterms and Maxterms

Each variable in a Boolean expression is a literal

Boolean variables can appear in normal (x) or complement form (x’)

Each AND combination of terms is a minterm Each OR combination of terms is a maxterm


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Minterms and Maxterms: Example

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Minterms

x y z Minterm0 0 0 x’y’z’ m0

0 0 1 x’y’z m1

…1 0 0 xy’z’ m4

…1 1 1 xyz m7

: Maxterms

x y z Maxterm0 0 0 x+y+z M0

0 0 1 x+y+z’ M1

…1 0 0 x’+y+z M4

…1 1 1 x’+y’+z’ M7


Functions with Minterms Minterm number same as row position

in truth table (starting from top from 0) Shorthand way to represent functions

x00001111

y00110011

z01010101

G00010011


G = m7 + m6 + m3 = Σ(3, 6, 7)


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Complementing Functions

x00001111

y00110011

z01010101

G00010011


G’ = (xyz + xyz’ + x’yz)’

G’11101100

Can we find a simpler representation?


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Conversion Between Canonical Forms

Easy to convert between minterm and maxterm representations

For maxterm representation, select rows with 0’s

x00001111

y00110011

z01010101

G00010011


G = m7 + m6 + m3 = Σ(3, 6, 7)

G = M0M1M2M4M5 = Π(0,1,2,4,5)

G = (x+y+z)(x+y+z’)(x+y’+z)(x’+y+z)(x’+y+z’)


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Karnaugh Maps

A Karnaugh map is a graphical tool for assisting in the general simplification procedure.

Two variable maps.

Three variable maps.

0A

1 11

00 01

0

1

BC

01 1

111 10

F=AB’C’ +AB C +ABC +ABC + A’B’C + A’BC’

0A

1 11

B0 1

0

1

F=AB +AB +AB

A B C F0 0 0 00 0 1 10 1 0 10 1 1 01 0 0 11 0 1 11 1 0 11 1 1 1

+


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Rules for K-MapsWe can reduce functions by circling 1’s in the

K-mapEach circle represents minterm reductionFollowing circling, we get minimized and-or

form.Rules to considerEvery cell containing a 1 must be included at

least once.The largest possible “power of 2 rectangle”

must be enclosed.The 1’s must be enclosed in the smallest

possible number of rectangles.Saras M. Srivastava PGT CS, KV IISc,

Bangalore 12

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Karnaugh Maps Two variable maps.

0A

1 11

B0 1

0

1 F=A+B

Three variable maps.

F=A+B C +BC 0

A

1 11

00 01

0

1

BC

01 1

111 10

F=AB +AB +AB

F=AB’C’ +AB C +ABC +ABC + A’B’C + A’BC’


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Karnaugh MapsNumbering scheme based on Gray–code

e.g., 00, 01, 11, 10 Only a single bit changes in code for

adjacent map cells00 01

ABC

0

1

11 10

C

B

A

F(A,B,C) = m(0,4,5,7)

G(A,B,C) = 0 0

0 0

1 1

1 1C

B

A

1 0

0 0

0 1

1 1C

B

A

A

= AC + B’C’


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Karnaugh Maps for Four Inputs

Represent functions of 4 inputs with 16 minterms

Use same rules developed for 3-input functions


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Karnaugh map: 4-variable example

F(A,B,C,D) =

m(0,2,5,8,9,10,11,12,13,14,15)F = C + B’D’+ A’BD

D

A

B

1 0

0 1

0 1

0 0

1 1

1 1

1 1

1 1C

AB

CD


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Karnaugh maps: Don’t cares

f(A,B,C,D) = m(1,3,5,7,9) + d(6,12,13) without don't cares

f = + C’D

0 0

1 1

X 0

X 1D

A

1 1

0 X

0 0

0 0

B

C

A’D

CDAB

00

01

11

10

00 01 11 10

C f0 00 11 01 10 00 11 X100110011

D0101010101010101

10100XX00

A0000000011111111

+

B0000111100001111

+


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Thank

YouSaras M. Srivastava PGT CS, KV IISc,

Bangalore 12

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boolean algebra saras m. srivastava pgt (computer science) kendriya vidyalaya, iisc bangalore –...

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