IDEAL EYES BUSINESS COLLEGE

APRESENTATION

ON BOOLEAN ALGEBRA

&LOGIC GATE

PRESENTED TO:- IEBC PRESENTED BY:- VIVEK KUMAR

CONTENT1. INTRODUCTION2. BOOLEAN LOGIC OPERATION3. LAWS& RULES OF BOOLEAN ALGEBRA4. DE MORGAN’S THEOREMS5. IMPLICATIONS OF DE MORGAN’S

THEOREMS6. COMBINATIONAL LOGIC7. KARNAUGH MAPS8. LOGIC GATE

INTRODUCTION• 1854: Logical algebra was published by George

Boole known today as “Boolean Algebra”• It’s a convenient way and systematic way of

expressing and analyzing the operation of logic circuits.

• 1938: Claude Shannon was the first to apply Boole’s work to the analysis and design of logic circuits.

• A Boolean algebra value can be either true or false.• Digital logic uses 1 to represent true and 0 to

represent false.

BOOLEAN LOGIC OPERATIONAND OPERATIONOR OPERATIONNOT (COMPLEMENTATION ) OPERATION

AND OPERATION• It is a two variables cases.• It is written as Y=A.B.• Dot (.) symbol is the common symbol of AND

gate.• When both input are 1 then output is also 1.• When both and also at list any input 0 then

output is also 0.• We can also write as Y=AB.

AND OPERATION TRUTH TABLE

OR OPERATION• It is also two variables case.• It is written as Y=A+B.• Plus (+) symbol is the common symbol of OR

gate.• When both input are 0 then output is also 0.• When both input 1 and at list any input 1

then output is also 1.

OR OPERATION TRUTH TABLE

NOT (COMPLEMENTATION ) OPERATION

•It is one variable case.•It has only one input.•It change any input to it’s compliment.•As like 1 to 0 & 0 to 1.•It is also written as A=A.•It is also called inverter.

LAWS & RULES OF BOOLEAN ALGEBRA

OPERATIONS WITH 0 AND 1:• 1. X + 0 = X 1D. X • 1 = X• 2. X + 1 = 1 2D. X • 0 = 0

• IDEMPOTENT LAWS• 3. X + X = X 3D. X • X = X

CONTINUE

LAWS & RULES OF BOOLEAN ALGEBRA

• 4. ( X' ) ' = X• LAWS OF COMPLEMENTARITY:• 5. X + X' = 1 5D. X • X' = 0

• COMMUTATIVE LAWS:• 6. X + Y = Y + X 6D. X • Y = Y • X

CONTINUE

• COMMUTATIVE LAWS:• 6. X + Y = Y + X 6D. X • Y = Y • X• ASSOCIATIVE LAWS: • 7. (X + Y) + Z = X + (Y + Z) 7D. (XY)Z = X(YZ) = XYZ• DISTRIBUTIVE LAWS:• 8. X( Y + Z ) = XY + XZ 8D. X + YZ = ( X + Y ) ( X +

Z )

LAWS & RULES OF BOOLEAN ALGEBRA

CONTINUE

LAWS & RULES OF BOOLEAN ALGEBRA

• SIMPLIFICATION THEOREMS:• 9. X Y + X Y' = X 9D. ( X + Y ) ( X + Y' ) = X• 10. X + XY = X 10D. X ( X + Y ) = X• 11. ( X + Y' ) Y = XY 11D. XY' + Y = X + Y• DEMORGAN’S LAWS:• 12. ( X + Y + Z + … )' = X'Y'Z'… 12D. (X Y Z …)' = X' + Y' + Z' + …• 13. [ f ( X1, X2, … XN, 0, 1, +, • ) ]' = f ( X1', X2', … XN', 1, 0, •, + )

CONTINUE

IMPLIMANTATION OF DE MORGAN’S THEOREMS

THEOREM 1A+B = A.B

AB

Y=A+BTHEOREM 2

AB

Y=A.B

DE MORGAN’S THEOREMS• As An Example, We Prove De Morgan’s Laws.

COMBINATIONAL LOGIC SOME OF PRODUCT (SOP) PRODUCT OF SOMS (POS)HOW TO CHANGE SOP TO POS & POS TO SOPCANONICAL FORMS

SOME OF PRODUCT (SOP)

• When two or more product terms are summed by Boolean addition,

• the resulting expression is a sum-of-products (SOP). Some examples are:

• AB + ABC• ABC + CDE + BCD• AB + BCD + AC• Also, an SOP expression can contain a single-

variable term, as in• A + ABC + BCD..

SOME OF PRODUCT (SOP)

• Example• Convert each of the following Boolean expressions

to SOP form:• (a) AB + B(CD + EF)

PRODUCT OF SOMS (POS)

When two or more sum terms are multiplied the resulting expression is a product-of-sums (POS).

Some examples are:-1 (A + B)(B + C + D)(A + C).2 (A + B + C)( C + D + E)(B + C + D)3 (A + B)(A + B + C)(A + C)

PRODUCT OF SUMS (POS)

(A+B)(B+C+D)(A+C)

HOW TO CHANGE SOP TO POS & POS TO SOP

• SOP TO POSEX:- AB + B(CD + EF)

Every (+) Sign Change Into( *) & Every * Sign Change In to (+) Sign.

Result Will Be(A+B)(B+C+D)(B+E+F)

HOW TO CHANGE SOP TO POS & POS TO SOP

• POS TO SOPEx:- (A+B)(B+C+D)(B+E+F)

Every (*) Sign Change Into( +) & Every (+)Sign Change In Yo (*) Sign.

Result Will BeAB + BCD + BEF

CANONICAL FORMS1 To Place A SOP Equation Into Canonical From Using

Boolean Algebra We Do The Following. Identify The Missing Variable In Each AND Terms.AND the missing terms and its complement with the

original AND term AB(C+C) because C+C =1,the original AND term value is not changed.

Expand the term by application of the proparty of the distribution, ABC+ABC

CANONICAL FORMS

2. To Place A POS Equation Into Canonical From Using Boolean Algebra We Do The Following.

Identify The Missing Variable In Each OR Terms.OR the missing terms and its complement with the

original OR term A+B+CC because CC =0,the original OR term value is not changed.

Expand the term by application of the proparty of the distribution, (A+B+C)(A+B+C).

CANONICAL FORMS

EX:- Convert A+B To Minterms.Solution:- A+B = A.1 + B.1

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

=AB+AB+BA+BA

minterms Y = A+B = AB+AB+BA

maxterms Y = A+B = (A+B)(A+B)(B+A)

K-MAPS INTRODUCTION

A Karnaugh map provides a systematic method for simplifying Booleanexpressions and, if properly used, will produce the simplest SOP or POSexpression possible, known as the minimum expression & maximum expression.

K-MAPS INTRODUCTIONNumber cells in k-maps depends upon thenumber of variables of boolean expression. K-maps can be used for any number of variables.But it is used upto six variables beyond which itis not very convenient,

1. 2-variable map contains 4 cells.2. 3-variable map contains 8 cells.3. 4-variable map contains 16 cells.4. n-variable map contains 2 on power n cells.

LOGIC GATEAND GATEOR GATENOT GATENAND GATENOR GATE EX-OR GATE EX-NOR GATE TRUTH TABLE LOGIC DIGRAM

AND FUNCTIONOutput Y is TRUE if inputs A ANDB are TRUE, else it is FALSE.

Logic Symbol

Text Description

Truth Table

Boolean Expression

AND

A

B

Y

INPUTS OUTPUT

A B Y

0 0 0

0 1 0

1 0 0

1 1 1 AND Gate Truth Table

Y = A x B = A • B = AB

AND Symbol

OR FUNCTIONOutput Y Is TRUE If Input A OR B Is TRUE or

both are TURE, Else It Is FALSE.

Logic Symbol

Text Description

Truth Table

Boolean Expression Y = A + B

OR Symbol

A

BYOR

INPUTS OUTPUT

A B Y

0 0 0

0 1 1

1 0 1

1 1 1 OR Gate Truth Table

NOT FUNCTION (INVERTER)Output Y Is TRUE If Input A Is FALSE, Else It Is

FALSE. Y Is The Inverse Of A.

Logic Symbol

Text Description

Truth Table

Boolean Expression

INPUT OUTPUT

A Y

0 1

1 0 NOT Gate Truth Table

A YNOT

Y = A

NAND FUNCTIONOutput Y is FALSE if inputs A AND B are TRUE,

else it is TRUE.

Logic Symbol

Text Description

Truth Table

Boolean Expression

A

BYNAND

A bubble is an inverter

This is an AND Gate with an inverted output

INPUTS OUTPUT

A B Y

0 0 1

0 1 1

1 0 1

1 1 0 NAND Gate Truth Table

Y=AB

NOR FUNCTIONOutput Y is FALSE if input A OR B is TRUE, or

both are TURE, else it is TRUE.

Logic Symbol

Text Description

Truth Table

Boolean Expression

A

BYNOR

A bubble is an inverter.

This is an OR Gate with its output inverted.

INPUTS OUTPUT

A B Y

0 0 1

0 1 0

1 0 0

1 1 0 NOR Gate Truth Table

Y =A+B