Theory of Consumer Behaviour Prof. Ravikesh Srivastava Lecture-7.
Theory Lecture 7
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Transcript of Theory Lecture 7
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MCT-212: DIGITAL LOGIC
DESIGN
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EXCLUSIVE OR GATE
The output of theXOR
gate is high whenever the two
inputs are different.
A B X = A B
0 0 0
1 0 1
0 1 1
1 1 0
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EXCLUSIVE NOR GATE
The output of the XNOR gate is high whenever the twoinputs are identical
A B X =
0 0 0
1 0 1
0 1 1
1 1 0
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BOOLEAN ALGEBRA AND LOGIC
DESIGN
The need forBoolean Algebra for logic design?
A B X
0 0 1
1 0 0
0 1 1
1 1 1
= + +
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BOOLEAN ALGEBRA AND LOGIC
DESIGN
The need forBoolean Algebra for logic design?
A B X
0 0 1
1 0 0
0 1 1
1 1 1
= + +
= +
= +
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BOOLEAN ALGEBRAzero one zero one zero
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BOOLEAN ALGEBRA
Mathematics of digital systems.
Important in the analysis of digital circuits.
Binary Variable : A symbol (letter) used to
represent a logic quantity.
Example - binary variables and their values:
A = 1
Z = 0
Complement : Inverse of a variable.
A = 1 , A= 0 / (A = 0)
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BOOLEAN ADDITION
A literal is a variable or its complement.
The addition is equivalent to OR operation.
The sum of terms is 1, if one ore more literals
are 1.
The sum is zero is each literal is 0.
Determine the value of A, B, C that will
make the sum term 0. + + = ?A = 0, B = 1, C = 0.
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BOOLEAN MULTIPLICATION
In Boolean algebra, multiplication is equivalent toAND operation.
The product will be 1 only if all of the literals are
1.What are the values of A, B, C that will
make the product term . . =
A = 1, B = 0, C = 1
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BOOLEAN OPERATIONS
Addition
0 + 0 = 0
1 + 0 = 1
0 + 1 = 1
1 + 1 = 1
Multiplication
0 * 0 = 0
1 * 0 = 0
0 * 1 = 0
1 * 1 = 1
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LAWS AND RULES OFBOOLEAN ALGEBRAWhat rules?
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LAWS AND RULES OF BOOLEAN
ALGEBRA
Copyright 2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458All rights reserved
Floyd
Digital Fundamentals, 9/e
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COMMUTATIVE LAWS
Applied to addition and multiplication.
Commutative law for addition: In terms of the
result, the order in which variables are ORed
makes no difference.
+ = +
Commutative law for Multiplication: In terms
of the result, the order in which variables are
ANDed makes no difference.
. = .
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ASSOCIATIVE LAWS
Also applied to addition and multiplication. Associative law for addition: When ORing
more than two variables, the result is the same
regardless of the grouping of the variables.
+ + = + +
Associative law for Multiplication: When
ANDing more than two variables, the result is the
same regardless of the grouping of the variables.
. . = . .
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ASSOCIATIVE LAWS
Associative law for addition: + + = + +
Associative law for Multiplication:
. . = . .
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VENN DIAGRAMS
The rules of Boolean algebra can be explained usingVenn Diagrams.
The variable A is shown as an area.
The rule + = can be illustrated easily with a
diagram. Add an overlapping area to represent thevariable B.
The overlap region
between A and B
represents AB
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VENN DIAGRAMS
The rules of Boolean algebra can be explained usingVenn Diagrams.
The diagram visually shows that + =
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VENN DIAGRAMS
What does this famous diagram say?? =D
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RULES OF BOOLEAN ALGEBRA
RULE 1: + =
A B X = A+B
0 0 0
1 0 10 1 1
1 1 1OR Truth Table
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RULES OF BOOLEAN ALGEBRA
RULE 2: + =
A B X = A+B
0 0 0
1 0 10 1 1
1 1 1OR Truth Table
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A B X = A.B
0 0 0
1 0 00 1 0
1 1 1
RULES OF BOOLEAN ALGEBRA
RULE 3:. =
AND Truth Table
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A B X = A.B
0 0 0
1 0 00 1 0
1 1 1
RULES OF BOOLEAN ALGEBRA
RULE 4:. =
AND Truth Table
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RULES OF BOOLEAN ALGEBRA
RULE 5: + =
A B X = A+B
0 0 0
1 0 10 1 1
1 1 1OR Truth Table
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RULES OF BOOLEAN ALGEBRA
RULE 6: + =
A B X = A+B
0 0 0
1 0 10 1 1
1 1 1OR Truth Table
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RULES OF BOOLEAN ALGEBRA
RULE 7:. =
A B X = A.B
0 0 0
1 0 00 1 0
1 1 1AND Truth Table
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RULES OF BOOLEAN ALGEBRA
RULE 9: =
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RULES OF BOOLEAN ALGEBRA
RULE 10: + =
Proof using Truth Table?
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RULES OF BOOLEAN ALGEBRA
RULE 11: + = +
Proof using Truth Table?
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RULES OF BOOLEAN ALGEBRA
RULE 12: ( + )( + ) = +
Proof using Venn Diagram?
Proof using Truth Table?
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RULES OF BOOLEAN ALGEBRA
RULE 12: ( + )( + ) = +
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DEMORGANS THEOREM
DeMorgans 1st Theorem: The complement ofa product of variables is equal to the sum of the
complemented variables.
= + A B +
0 0 1 1
1 0 1 1
0 1 1 1
1 1 0 0
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DEMORGANS THEOREM
DeMorgans 2st Theorem: The complement ofa sum of variables is equal to the product of the
complemented variables. + = .
Use DeMorgans Theorem to solve the following:
. +
A B + .
0 0 1 11 0 0 0
0 1 0 0
1 1 0 0
SIMPLIFICATION USING BOOLEAN
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SIMPLIFICATION USING BOOLEAN
ALGEBRA
Dont look here, simplify the expression writtenon the board..!
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STANDARD FORMS OF
BOOLEAN EXPRESSIONSBasssss hogyi hai ab to :(
STANDARD EXPRESSIONS FROM
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STANDARD EXPRESSIONS FROM
TRUTH TABLE
Truth tables show allpossible inputs of a
function and the values
that the output takes for
all those inputs.
Given n inputs, there are
2 possible input
combinations.
A B C F(A,B,C)
0 0 0 0
0 0 1 0
0 1 0 1
0 1 1 1
1 0 0 0
1 0 1 0
1 1 0 1
1 1 1 0
STANDARD EXPRESSIONS FROM
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STANDARD EXPRESSIONS FROM
TRUTH TABLE
Domain of a Boolean expression: The set ofvariables contained in the expression in either
uncomplemented and complemented form
The sum-of-product (SOP) form
X = AB + CD + EF
Domain: A,B,C,D,E,F
The product of sum (POS) form = ( + + )( + + )
In both forms, the over bar cannot extend over more than
one variable.
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SUM OF PRODUCTS STANDARD FORM
In SOP standard form, every variable in thedomain must appear in each term. This form is
useful for constructing truth tables or for
implementing logic in PLDs.
You can expand a nonstandard term to standard
form by multiplying the term by a term consisting
of the sum of the missing variable and its
complement.
TRUTH TABLES TO MIN TERM
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TRUTH TABLES TO MIN-TERM
EXPRESSIONS
NOT, AND , and OR
How would you create a min-term
expression from a truth table?
The need for making standardexpressions?
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PRODUCT OF SUMS FORM
Similarly, a product-of-sums is formed by the product ofsums in which all the sums are formed by single variables
only. Examples:
(A + B)(C + D + E)(A + C + E)
(A + B)(C + D +E)F
Each of the sums in the product-of-sums form is called a
maxterm. Thus the form is also called product-of-
maxterms. The following expressions are not sum-of-products:
(A+B)CD + EF
A + B + C + DE
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PRODUCT OF SUMS STANDARD FORM
In POS standard form, every variable in thedomain must appear in each sum term of the
expression.
You can expand a nonstandard POS expressionto standard form by adding the product of the
missing variable and its complement.
Making POS from truth table?
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INTEGRATED CIRCUITGATES
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DIGITAL LOGIC FAMILIES
TTL Transistor Transistor Logic ECL Emitter Coupled Logic
MOS Metal Oxide Semiconductor
CMOS Complementary Metal Oxide
Semiconductor
Two major Fixed Function Logic families are
TTLand CMOS.
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FIXED FUNCTION LOGIC
Packaging for fixed function logic is shown:
Copyright 2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458All rights reserved
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Some common pin configuration are shown:
Copyright 2006 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved
PERFORMANCE CHARACTERISTICS
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PERFORMANCE CHARACTERISTICS
AND PARAMETERS
Propagation delay Time DC Supply Voltage (VCC)
Power Dissipation
Input and Output Logic Levels Speed-Power product
Fan-Out and Loading
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PROPAGATION DELAY
Delay in the output change after the inputchanges.
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FAN-OUT AND LOADING
Maximum number of ICs that can be driven fromthe output of a single IC.
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ANY QUESTIONS?
Anyone willing to present?
Time allowed : 5 mins
Topic : Any
Bonus Points : +3
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REFERENCES
Chapter no 1: Binary Systems Digi tal Lo gic Designby Morris Mano
Chapter no 1: Digital Concepts
Digi tal Fundamentalsby Floyd