The document provides an overview of Boolean algebra and logic simplification. It covers topics such as Boolean variables that can take true/false or 1/0 values, basic logic gates like AND, OR, NOT, NAND and NOR gates, canonical forms including sum-of-products and product-of-sums, De Morgan's laws, and examples of simplifying Boolean expressions and implementing logic circuits.
Contents:
1.What is number system?
2.Conversions of number from one radix to another
3.Complements (1's, 2's, 9's, 10's)
4.Binary Arithmetic ( Addition, subtraction, multiplication, division)
It is the adder used to eliminate the wastage of time occur at each stage of parallel binary adder.In this , by using only carry input signal , we can calculate the the carry output without going to calculate carry at each stage.it is commonly used only for 4 bit addition because further calculation will be more complex.
In which i describe all the features of decoder. All the functionalities describe with the circuits and truth tables. So download and learn more about decoder. Decoder Full Presentation.
Contents:
1.What is number system?
2.Conversions of number from one radix to another
3.Complements (1's, 2's, 9's, 10's)
4.Binary Arithmetic ( Addition, subtraction, multiplication, division)
It is the adder used to eliminate the wastage of time occur at each stage of parallel binary adder.In this , by using only carry input signal , we can calculate the the carry output without going to calculate carry at each stage.it is commonly used only for 4 bit addition because further calculation will be more complex.
In which i describe all the features of decoder. All the functionalities describe with the circuits and truth tables. So download and learn more about decoder. Decoder Full Presentation.
Principles of Combinational Logic: Definition of combinational logic, canonical forms, Generation of switching equations from truth tables, Karnaugh maps-3,4,5 variables, Incompletely specified functions (Don‘t care terms) Simplifying Max term equations
This topic introduces the numbering systems: decimal, binary, octal and hexadecimal. The topic covers the conversion between numbering systems, binary arithmetic, one's complement, two's complement, signed number and coding system. This topic also covers the digital logic components.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Event Management System Vb Net Project Report.pdfKamal Acharya
In present era, the scopes of information technology growing with a very fast .We do not see any are untouched from this industry. The scope of information technology has become wider includes: Business and industry. Household Business, Communication, Education, Entertainment, Science, Medicine, Engineering, Distance Learning, Weather Forecasting. Carrier Searching and so on.
My project named “Event Management System” is software that store and maintained all events coordinated in college. It also helpful to print related reports. My project will help to record the events coordinated by faculties with their Name, Event subject, date & details in an efficient & effective ways.
In my system we have to make a system by which a user can record all events coordinated by a particular faculty. In our proposed system some more featured are added which differs it from the existing system such as security.
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
Contact with Dawood Bhai Just call on +92322-6382012 and we'll help you. We'll solve all your problems within 12 to 24 hours and with 101% guarantee and with astrology systematic. If you want to take any personal or professional advice then also you can call us on +92322-6382012 , ONLINE LOVE PROBLEM & Other all types of Daily Life Problem's.Then CALL or WHATSAPP us on +92322-6382012 and Get all these problems solutions here by Amil Baba DAWOOD BANGALI
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Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
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Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
2. Slide 2
Boolean Algebra
• Boolean algebra is a mathematical system for the
manipulation of variables that can have one of two values.
– In formal logic, these values are “true” and “false.”
– In digital systems, these values are “on” and “off,” 1 and
0, or “high” and “low.”
• When we learned numbers like 1, 2, 3, we also then learned
how to add, multiply, etc. with them. Boolean Algebra
covers operations that we can do with 0’s and 1’s.
Computers do these operations ALL THE TIME and they
are basic building blocks of computation inside your
computer program.
3. Slide 3
How does Boolean Algebra fit into
the big picture?
• It is part of the Combinational Logic topics (memoryless)
– Different from the Sequential logic topics (can store
information)
• Learning Axioms and theorems of Boolean algebra
• Allows you to design logic functions
• Allows you to know how to combine different logic
gates
• Allows you to simplify or optimize on the complex
operations
4. Slide 4
NOT Gate -- Inverter
X Y
0
1
1
0
X Y
Y
NOT
X Y
Y = ~X
NOT
• The NOT operation is most often designated by an
overbar. It is sometimes indicated by a prime mark ( ‘ ) or
an “elbow” ().
Basic Logic Gates
5. Slide 5
AND Gate
AND (Multiplication)
X
Y
Z
Z = X . Y
X Y Z
0 0 0
0 1 0
1 0 0
1 1 1
6. Slide 6
OR Gate
OR (Addition)
X
Y
Z
Z = X + Y
X Y Z
0 0 0
0 1 1
1 0 1
1 1 1
8. Slide 8
NAND Gate
X
Y
X
Y
Z
Z
Z = ~(X . Y) Z = ~X + ~Y
=
X Y W Z
0 0 0 1
0 1 0 1
1 0 0 1
1 1 1 0
X Y ~X ~Y Z
0 0 1 1 1
0 1 1 0 1
1 0 0 1 1
1 1 0 0 0
W
10. Slide 10
NOR Gate
X
Y
Z
Z = ~(X + Y)
X Y Z
0 0 1
0 1 0
1 0 0
1 1 0
X
Y
Z
Z = ~X . ~Y
X Y ~X ~Y Z
0 0 1 1 1
0 1 1 0 0
1 0 0 1 0
1 1 0 0 0
11. Slide 11
Exclusive-OR Gate
X Y ZXOR
X
Y
Z 0 0 0
0 1 1
1 0 1
1 1 0
xor(Z,X,Y)
• The output of an XOR gate is true (1) only when exactly one of its inputs
is true (1). If both of an XOR gate's inputs are false (0), or if both of its
inputs are true (1), then the output of the XOR gate is false (0).
12. Slide 12
Exclusive-NOR Gate
X Y ZXNOR
X
Y
Z 0 0 1
0 1 0
1 0 0
1 1 1
xnor(Z,X,Y)
• The output of an XNOR gate is true (1) when all of its inputs are true (1)
or when all of its inputs are false (0). If some of its inputs are true (1) and
others are false (0), then the output of the XNOR gate is false(0).
17. Slide 17
Some notation
• Priorities:
1.Parentheses
2.NOT > AND > XOR > OR
• Variables and their complements are
sometimes called literals
A B+C =((A) B)+C
18. Slide 18
Laws of Boolean Algebra
• Commutative Law of Addition:
A + B = B + A
19. Slide 19
Laws of Boolean Algebra
• Commutative Law of Multiplication:
A * B = B * A
20. Slide 20
Laws of Boolean Algebra
• Associative Law of Addition:
A + (B + C) = (A + B) + C
21. Slide 21
Laws of Boolean Algebra
• Associative Law of Multiplication:
A * (B * C) = (A * B) * C
22. Slide 22
Laws of Boolean Algebra
• Distributive Law:
A(B + C) = AB + AC
23. Slide 23
Rules of Boolean Algebra
• Rule 1 IDENTITY w.r.t ADDITION
OR Truth Table
24. Slide 24
Rules of Boolean Algebra
• Rule 2 NULL w.r.t ADDITION
OR Truth Table
25. Slide 25
Rules of Boolean Algebra
• Rule 3 NULL w.r.t MULTIPLICATION
AND Truth Table
26. Slide 26
Rules of Boolean Algebra
• Rule 4 IDENTITY w.r.t MULTIPLICATION
AND Truth Table
27. Slide 27
Rules of Boolean Algebra
• Rule 5 IDEMPOTENT w.r.t ADDITION
OR Truth Table
28. Slide 28
Rules of Boolean Algebra
• Rule 6 COMPLEMENTARITY w.r.t
ADDITION
OR Truth Table
29. Slide 29
Rules of Boolean Algebra
• Rule 7 IDEMPOTENT w.r.t
MULTIPLICATION
AND Truth Table
30. Slide 30
Rules of Boolean Algebra
• Rule 8 COMPLEMENTARITY w.r.t
MULTIPLICATION
AND Truth Table
43. Slide 43
Canonical forms
• Canonical forms
– Standard forms for Boolean expressions
– Derived from truth table
– Generally not the simplest forms (can be
minimized)
• Two canonical forms
– Sum-of-products (minterms)
– Product-of-sums (maxterms)
44. Slide 44
Sum-of-products (SOP)
• Also called disjunctive normal form (DNF)
or minterm expansion
A B C F F'
0 0 0 0 1
0 0 1 1 0
0 1 0 0 1
0 1 1 1 0
1 0 0 0 1
1 0 1 1 0
1 1 0 1 0
1 1 1 1 0
001 011 101 110 111
F = A'B'C + A'BC + AB'C + ABC' + ABC
F' = A'B'C' + A'BC' + AB'C'
minterm
0= ‘
NOTICE 1 TO SOLVE
45. Slide 45
Minterms
• Variables appear exactly once in each
minterm in true or inverted form (but not
both)
short-hand notation
A B C minterms
0 0 0 A'B'C' m0
0 0 1 A'B'C m1
0 1 0 A'BC' m2
0 1 1 A'BC m3
1 0 0 AB'C' m4
1 0 1 AB'C m5
1 1 0 ABC' m6
1 1 1 ABC m7
F in canonical form:
F(A,B,C) = m(1,3,5,6,7)
= m1 + m3 + m5 + m6 + m7
= A'B'C+A'BC+AB'C+ABC'+ABC
46. Slide 46
Product-of-sums (POS)
• Also called conjunctive normal form (CNF)
or maxterm expansion
A B C F F'
0 0 0 0 1
0 0 1 1 0
0 1 0 0 1
0 1 1 1 0
1 0 0 0 1
1 0 1 1 0
1 1 0 1 0
1 1 1 1 0
000 010 100
F = (A + B + C) (A + B' + C) (A' + B + C)
F' = (A+B+C')(A+B'+C')(A'+B+C')(A'+B'+C)(A'+B'+C')
maxterm
1 = ‘
NOTICE 0 TO SOLVE
47. Slide 47
Maxterms
• Variables appear exactly once in each
maxterm in true or inverted form (but not
both)
A B C maxterms
0 0 0 A+B+C M0
0 0 1 A+B+C' M1
0 1 0 A+B'+C M2
0 1 1 A+B'+C' M3
1 0 0 A'+B+C M4
1 0 1 A'+B+C' M5
1 1 0 A'+B'+C M6
1 1 1 A'+B'+C' M7
short-hand notation
F in canonical form:
F(A,B,C) = M(0,2,4)
= M0 • M2 • M4
= (A+B+C)(A+B'+C)(A'+B+C)
49. Slide 49
Example
• Truth table for f1(a,b,c) at right
• The canonical sum-of-products form for
f1 is
f1(a,b,c) = m1 + m2 + m4 + m6
= a’b’c + a’bc’ + ab’c’ + abc’
• The canonical product-of-sums form for
f1 is
f1(a,b,c) = M0 • M3 • M5 • M7
= (a+b+c)•(a+b’+c’)•
(a’+b+c’)•(a’+b’+c’).
a b c f1
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 0
1 0 0 1
1 0 1 0
1 1 0 1
1 1 1 0
50. Slide 50
Example 2
Design the minimum-cost product-of-
sums and sum-of-product expression for
the function
f(x1, x2, x3) = Σ m(0, 2, 4, 5, 6, 7)
52. Slide 52
Minterms and Maxterms
(with three variables)
The function is
1 for these rows
53. Slide 53
Minterms and Maxterms
(with three variables)
The function is
1 for these rows
The function is
0 for these rows
54. Slide 54
From SOP to POS and back
• Minterm to maxterm
– Use maxterms that aren’t in minterm
expansion
– F(A,B,C) = m(1,3,5,6,7) = M(0,2,4)
• Maxterm to minterm
– Use minterms that aren’t in maxterm
expansion
– F(A,B,C) = M(0,2,4) = m(1,3,5,6,7)
55. Slide 55
From SOP to POS and back
• Minterm of F to minterm of F'
– Use minterms that don’t appear
– F(A,B,C) = m(1,3,5,6,7) F' = m(0,2,4)
• Maxterm of F to maxterm of F'
– Use maxterms that don’t appear
– F(A,B,C) = M(0,2,4)F' = M(1,3,5,6,7)
56. Slide 56
SOP, POS, and DeMorgan's
• Product-of-sums
– F' = (A+B+C')(A+B'+C')(A'+B+C')(A'+B'+C')
• Apply DeMorgan's to get SOP
– (F')' = ((A+B+C')(A+B'+C')(A'+B+C')(A'+B'+C'))'
– F = A'B'C + A'BC + AB'C + ABC
57. Slide 57
SOP, POS, and DeMorgan's
• Sum-of-products
– F' = A'B'C' + A'BC' + AB'C'
• Apply DeMorgan's to get POS
– (F')' = (A'B'C' + A'BC' + AB'C')'
– F = (A+B+C)(A+B'+C)(A'+B+C)
58. Slide 58
Conversion of SOP from standard to canonical form
• Expand non-canonical terms by inserting
equivalent of 1 in each missing variable x:
(x + x’) = 1
• Remove duplicate minterms
• f1(a,b,c) = a’b’c + bc’ + ac’
= a’b’c + (a+a’)bc’ + a(b+b’)c’
= a’b’c + abc’ + a’bc’ + abc’ + ab’c’
= a’b’c + abc’ + a’bc + ab’c’
59. Slide 59
Conversion of POS from standard to canonical form
• Expand noncanonical terms by adding 0 in terms
of missing variables (e.g., xx’ = 0) and using the
distributive law (e.g., A + (BC) = (A + B).(A + C))
• Remove duplicate maxterms
• f1(a,b,c) = (a+b+c)•(b’+c’)•(a’+c’)
= (a+b+c)•(aa’+b’+c’)•(a’+bb’+c’)
= (a+b+c)•(a+b’+c’)•(a’+b’+c’)•
(a’+b+c’)•(a’+b’+c’)
= (a+b+c)•(a+b’+c’)•(a’+b’+c’)•(a’+b+c’)
69. Slide 69
– Multiple NOR = a complement of OR gate
– Multiple NAND = a complement of AND
– The cascaded NAND operations = sum of products
– The cascaded NOR operations = product of sums
70. Slide 70
2-70
– The XOR and XNOR gates are commutative and
associative
– Multiple-input XOR gates are uncommon.
– XOR is an odd function: it is equal to 1 if the inputs
variables have an odd number of 1's
71. Slide 71
Two-Level NAND Gate Implementation - Example
F (X,Y,Z) = m(0,6)
1. Express F in SOP form:
F = X’Y’Z’ + XYZ’
2. Obtain the AND-OR implementation for F.
3. Add bubbles and inverters to transform
AND-OR to NAND-NAND gates.
76. Slide 76
1. The associative law for addition is normally written as
a. A + B = B + A
b. (A + B) + C = A + (B + C)
c. AB = BA
d. A + AB = A
2. The Boolean equation AB + AC = A(B+ C) illustrates
a. the distribution law
b. the commutative law
c. the associative law
d. DeMorgan’s theorem
77. Slide 77
3. The Boolean expression A . 1 is equal to
a. A
b. B
c. 0
d. 1
4. The Boolean expression A + 1 is equal to
a. A
b. B
c. 0
d. 1
78. Slide 78
5. The Boolean equation AB + AC = A(B+ C) illustrates
a. the distribution law
b. the commutative law
c. the associative law
d. DeMorgan’s theorem
6. A Boolean expression that is in standard SOP form is
a. the minimum logic expression
b. contains only one product term
c. has every variable in the domain in every term
d. none of the above