This document discusses Boolean algebra and logic gates. It begins with an introduction to binary logic and Boolean variables that can take on values of 0 or 1. It describes logical operators like AND, OR, and NOT. Boolean algebra provides a mathematical system for specifying and transforming logic functions. The document provides examples of Boolean functions and logic gates. It discusses topics like Boolean variables and values, Boolean functions, logical operators, Boolean arithmetic, theorems, and algebraic proofs. George Boole is credited with developing Boolean algebra. Truth tables and Karnaugh maps are shown as ways to analyze Boolean functions.
In this slide, the following topics are discussed. Radix number system, Binary number system, Octal, Hexadecimal, Octal to Binary, Binary to Octal, Hexadecimal to binary, Binary to Hexadecimal, BCD codes, Gray codes, one's complement, two's complement, signed magnitude number system, fixed point representation, floating point representation and their conversion.
In this slide, the following topics are discussed. Radix number system, Binary number system, Octal, Hexadecimal, Octal to Binary, Binary to Octal, Hexadecimal to binary, Binary to Hexadecimal, BCD codes, Gray codes, one's complement, two's complement, signed magnitude number system, fixed point representation, floating point representation and their conversion.
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Computer Architecture and Organization
V semester
Anna University
By
Babu M, Assistant Professor
Department of ECE
RMK College of Engineering and Technology
Chennai
Because binary logic is used in all of today´s digital computers and devices, the cost of the circuit that implement it is important factor addressed by designers- be they computer engineers, electrical engineers, or computer scientist.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Magnitude Comparator ppt..
You can watch my lectures at:
Digital electronics playlist in my youtube channel:
https://www.youtube.com/channel/UC_fItK7wBO6zdWHVPIYV8dQ?view_as=subscriber
My Website : https://easyninspire.blogspot.com/
Binary addition, Binary subtraction, Negative number representation, Subtraction using 1’s complement and 2’s complement, Binary multiplication and division, Arithmetic in octal, hexadecimal number system, BCD and Excess – 3 arithmetic
Audio Version available in YouTube Link : www.youtube.com/Aksharam
subscribe the channel
Computer Architecture and Organization
V semester
Anna University
By
Babu M, Assistant Professor
Department of ECE
RMK College of Engineering and Technology
Chennai
Because binary logic is used in all of today´s digital computers and devices, the cost of the circuit that implement it is important factor addressed by designers- be they computer engineers, electrical engineers, or computer scientist.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Magnitude Comparator ppt..
You can watch my lectures at:
Digital electronics playlist in my youtube channel:
https://www.youtube.com/channel/UC_fItK7wBO6zdWHVPIYV8dQ?view_as=subscriber
My Website : https://easyninspire.blogspot.com/
Binary addition, Binary subtraction, Negative number representation, Subtraction using 1’s complement and 2’s complement, Binary multiplication and division, Arithmetic in octal, hexadecimal number system, BCD and Excess – 3 arithmetic
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This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
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Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
2. Binary Logic and Gates
• Binary variables take on one of two values.
• Logical operators operate on binary values and
binary variables.
• Basic logical operators are the logic functions AND,
OR and NOT.
• Logic gates implement logic functions.
• Boolean Algebra: a useful mathematical system for
specifying and transforming logic functions.
• We study Boolean algebra as a foundation for
designing and analyzing digital systems!
Boolean Algebra and Logic
Gates 2
3. Boolean Variables and Values
A Boolean variable, usually denoted in
common letters, as a, b,..x. y etc can take
only two values 0 and 1.
So we can say x = 0 or x = 1 and y = 0 or y
= 1 but these are the ONLY values the
Boolean variables x and y can take.
3
4. Boolean functions
• A Boolean function F(x, y) takes Boolean
variables and outputs a Boolean value.
• For instance, we can have
F(x, y) = x + x.y where x and y are
Boolean variables.
• Again, F(x, y) = xy’ as taken from the Unit
notes
4
5. Logical Operators
• In the last slide, we had the Boolean
function F(x, y) = x + x.y
• Does the function expression involve
operators?
5
7. Logical Operators
• Logical operators operate on
binary values and binary variables.
• For instance 0.1 = 0 and 1 + 1 = 1
so we have to learn a new
arithmetic!
7
8. George Boole (1815-1864)
8
George Boole was
a largely self-taught
English
mathematician,
philosopher and
logician, most of
whose short career
was spent as the
first professor of
mathematics at
Queen's College,
Cork in Ireland.
10. Boolean Single Value Theorems
4–10
x 1 x + 1
0 1 1
1 1 1
We can prove the above assertions using
truth tables as follows:
How can you
interpret this
truth table?
11. 11
Absorption Law
Use a truth table to show that for two
Boolean variables x, y and z: x + xy = x
x y xy x + xy
0 0
0 1
1 0
1 1
12. 12
Absorption Law
Use a truth table to show that for two
Boolean variables x, y and z: x + xy = x
x y xy x + xy
0 0 0 0
0 1 0 0
1 0 0 1
1 1 1 1
15. Boolean Algebraic Proofs
• Our primary reason for doing proofs is to
learn:
– Careful and efficient use of the identities and
theorems of Boolean algebra, and
– How to choose the appropriate identity or
theorem to apply to make forward progress,
irrespective of the application.
Boolean Algebra and Logic
Gates 15
16. Boolean Algebra
16
Boolean algebra refers to symbolic manipulation of expressions made up of boolean
variables and boolean operators. The familiar identity, commutative, distributive, and
associative axioms from algebra define the axioms of Boolean algebra, along with the
two complementary axioms.
17. Use Boolean algebra to prove that: x + x · y = x
Proof Steps Justification
x + x · y
= x · 1 + x · y Identity element: x · 1 = x
= x · ( 1 + y) Distributive
= x · 1 1 + y = 1
= x Identity element
Boolean Algebra and Logic
Gates 17
19. Use Boolean algebra to show that (x + y) (x + z) = x + yz
19
Proof Steps Justification
(x + y)(x + z)
= x x + xz + yx + yz Expanding
= x + xz + yx + yz Idempotent Law of
Multiplication
= x + x (z + y) + yz Distributive Law
= x + yz Absorption Law
20. Use Boolean algebra to show that (x + y) (x + z) = x + yz
20
Proof Steps Justification
(x + y)(x + z)
= x + yz Absorption Law
21. Is it still true that: (x + y)(x + z) = x(x + z) + y (x + z)
21
x y z x + y x + z (x +y)(x + z) x(x + z) y(x + z) x(x+z)+y(x+z)
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
23. Boolean Functions
23
A Boolean function accepts binary inputs and outputs a single
binary value.
A Boolean function is a mathematical function that maps
arguments to a value where the allowable values of range (the
function arguments) and domain (the function value) are just
one of two values, 0 or 1.
We usually denote Boolean functions as F(x, y) where x and y
are Boolean variables. We use truth tables to evaluate the
functions.
https://introcs.cs.princeton.edu/java/71boolean/
24. Boolean function
• We can define a Boolean function:
F(x, y) = (x AND y) OR (NOT x AND NOT y)
• We write: F(x, y, z) = xy + x’y’
• and then write a “truth table” for it:
24
x y x’ y’ xy x’y’ F(x, y ,z) = xy + x’y’)
0 0 1 1 0 1 1
0 1 1 0 0 0 0
1 0 0 1 0 0 0
1 1 0 0 1 0 1
• Finally, use gates to implement the function: F(x, y) = xy + x’y
• We use two AND gates and one OR gate.
25. Boolean function
• Another example:
F(x, y, z) = (x AND y AND (NOT z))
OR (x AND (NOT y) AND z)
OR ((NOT x) AND Y AND Z)
OR xyz
• We write this as:
F(x, y, z) = xyz’ + xy’z + x’yz + xyz
25
• Can this function be simplified?
• We can use the Boolean Expression Calculator to help with
this…
27. Logicly [logicly.ly]
The application “Logicly” can be used to simulate the circuit as
shown:
You can take a free trial and then either download it on your
computer to use it or use it online.
27
31. Minterms
A minterm, denoted by mi, where 0
≤ I < 2n, is a product (AND) of the n
variables in which each variable is
complemented if the value
assigned to it is 0, and
uncomplemented if it is 1.
We speak of the 1-minterms =
minterms for which the function F =
1 and 0-minterms, the minterms for
which the function F = 0.
Any Boolean function can be
expressed as a sum (OR) of its 1-
minterms. We write F(x, y z) = ∑(3, 5, 6, 7)
33. Boolean Function F(x, y, z) = x + yz
33
The truth table can show us all the outputs of F(x, y, z) for various
values of the inputs:.
x y z yz x + yz
0 0 0 0 m0 = 0
0 0 1 0 m1 = 0
0 1 0 0 m2 = 0
0 1 1 1 m3 = 1
1 0 0 0 m4 = 1
1 0 1 0 m5 = 1
1 1 0 0 m6 = 1
1 1 1 1 m7 = 1
So here we have F(x, y, z) = x’yz + xy’z’ + xy’z + xyz’ +xyz
37. 37
F(x, y, z) = x’yz + xy’z’ + xy’z + xyz’ +xyz = x + yz
How many gates does each of the expressions use?
38. Recall: (x + y) (x + z) = x + yz
and now we have
x + yz = x’yz + xy’z’ + xy’z + xyz’ +xyz
Is there a way to prove all this algebraically…
38
Proof Steps Justification
x’yz + xy’z’ + xy’z + xyz’ +xyz
= x’yz + xy’z’ + xy’z + xyz’ +xyz + xyz Idempotent
= (x’ + x) yz + xy (z + z’) + xy’(z + z’) Distributive
= yz + xy + xy’ Complement law
= x(y + y’) + yz Distributive Law
= x + yz Absorption law
39. Show that: x’yz + xy’z + xyz’ +xyz = xy + yz + + yz
39
Proof Steps Justification
x’yz + xy’z + xyz’ +xyz
= x’yz + xy’z + xyz’ +xyz +xyz + xyz Idempotent
= xy(z + z’) + xz (y + y’) + yz (x + x’) Distributive
= xy + yz + xz Complement law
40. We have seen that (x + y) (x + z) = x + yz
40
(x + y) (x + z) is a Product of Sums (POS)
expression
snd x + yz is a Sum of Products (SOP)
So F(x, y z) = (x + y)(x + z) = x + yz
Ie. The same function can be represented in
SOP or POS format.
41. Express the function F(x, y, z) = x + yz as a
POS expression.
41
Proof Steps Justification
x + yz
= x + x (z + y) + yz
= x + xz + xy + yz
= xx + xz + xy + yz
= (x + y)(x + z) as per required.
42. Definitions
42
The boolean expression that we construct is
known as the sum-of-products
representation is also known as the
disjunctive normal form of the function.
A boolean expression consisting entirely
either of minterm or maxterm is called
canonical expression.
43. 43
Express the Boolean function F(a, b, c) = (b + ac’)’(ab’ + c) in
Sum-of-Products form using Boolean algebra laws and
theorems. Express in the minimal expression.
F(a, b, c) = (b + ac’)’(ab’ + c)
= (b’)(ac’)’(ab’ + c) De Morgan’s Law
= b’(a’ + c)(ab’ + c) De Morgan’s law
= (a’b’ + b’c)(ab’ + c) Distributive Law
= a’b’ab’ + a’b’c + b’cab’ + b’cc
= a’b’c + ab’c + b’c
= b’c(a’ + a + 1)
= b’c
44. 44
We had earlier expressed the Boolean function F(a, b, c) = (b + ac’)’(ab’ +
c) in Sum-of-Products form using Boolean algebra laws and theorems.
Express this in canonical form.
F(a, b, c) = (b + ac’)’(ab’ + c)
a b c b’ c’ ac’ b + ac’ (b + ac’)’ ab’ ab’ + c F(a, b, c)
0 0 0 1 1 0 0 1 0 0 0
0 0 1 1 0 0 0 1 0 1 1
0 1 0 0 1 0 1 0 0 0 0
0 1 1 0 0 0 1 0 0 1 0
1 0 0 1 1 1 1 0 1 1 0
1 0 1 1 0 0 0 1 1 1 1
1 1 0 0 1 1 1 0 0 0 0
1 1 1 0 0 0 1 0 0 1 0
F(a, b, c) = a’b’c + ab’c
45. Canonical Form
• Express in canonical form…F(X, Y, Z) = X Y + Y ‘Z
X Y Z X Y Y’ Y ‘Z F = X Y + Y ‘Z
0 0 0 0 1 0 0
0 0 1 0 1 1 1
0 1 0 0 0 0 0
0 1 1 0 0 0 0
1 0 0 0 1 0 0
1 0 1 0 1 1 1
1 1 0 1 0 0 1
1 1 1 1 0 0 1
47. Claude Shannon
47
Shannon's work became the
foundation of digital circuit
design, as it became widely
known in the electrical
engineering community during
and after World War II. The
theoretical rigor of Shannon's
work superseded the ad hoc
methods that had prevailed
previously..
48. Boolean Algebra Simplification
Boolean Algebra and Logic
Gates 48
11 1 1
11 1 0
11 0 1
11 0 0
00 1 1
00 1 0
10 0 1
00 0 0
X Y Z F(x, y, z) From the minterms expansion, we have:
F(x, y, z) = x’y’z + xy’z’ +xy’z + xyz’ + xyz
= (x’ + x)y’z + xz’(y’ + y) + xyz
= y’z + xz’y’ + xz’y + xyz
= y’z + x(yz + yz’ + z’y’)
= y’z + x(yz + yz’ + (yz)’)
= y’z + x(1 + yz’)
= x + y’z
We can use the Boolean algebra laws to simplify a function. What is the
formula for F9x, y, z) from the following?
49. From the Boolean expression to the logic gate diagram…
Boolean Algebra and Logic
Gates 49
11 1 1
11 1 0
11 0 1
11 0 0
00 1 1
00 1 0
10 0 1
00 0 0
X Y Z F(x, y, z) F(x, y, z) = x’y’z + xy’z’ +xy’z + xyz’ + xyz
= (x’ + x)y’z + xz’(y’ + y) + xyz
= y’z + xz’y’ + xz’y + xyz
= y’z + x(yz + yz’ + z’y’)
= y’z + x(yz + yz’ + (yz)’)
= y’z + x(1 + yz’)
= x + y’z
We can use the Boolean algebra laws to simplify a function. What is the
formula for F9x, y, z) from the following?
X
Y F
Z