BOOLEAN ALGEBRA & LOGIC GATE
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Transcript of BOOLEAN ALGEBRA & LOGIC GATE
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