The document discusses different number systems including decimal, binary, octal, and hexadecimal. It explains how each system uses a different set of digits to represent values and how numbers are expressed as a sum of weighted place values. Conversion between different number systems like binary to decimal is demonstrated. Gray codes and binary codes for representing decimal digits are also covered. Boolean algebra concepts such as logic gates, truth tables, and identities are defined.
This document discusses number representation systems used in computers, including binary, decimal, octal, and hexadecimal. It provides examples of converting between these different bases. Specifically, it covers:
1) Converting between decimal, binary, octal, and hexadecimal using positional notation.
2) Signed integer representation in binary, including sign-magnitude, one's complement, and two's complement. Examples are given of converting positive and negative decimals to these binary representations.
3) Storing integer, character, and floating point numbers in binary. Twos complement is described as the most common method for signed integer representation.
This document discusses number representation systems used in computers, including binary, decimal, octal, and hexadecimal. It provides examples of converting between these different bases. Specifically, it covers:
1) Converting between decimal, binary, octal, and hexadecimal using positional notation and place values.
2) Representing signed integers in binary using ones' complement and twos' complement notation.
3) Tables for converting binary numbers to octal and hexadecimal using place values of each base.
4) Examples of converting values between the different number bases both manually and using the provided conversion tables.
This document provides an introduction to number systems and binary logic gates. It discusses radix numbers including binary, octal, decimal, and hexadecimal. Methods for converting between bases are presented, along with examples. Binary arithmetic operations like addition, subtraction, and multiplication are also covered. The main logic gates - NOT, AND, OR, XOR, NAND, NOR, and XNOR - are defined through their truth tables. Circuit design is discussed, where combinations of gates can represent Boolean expressions. Examples of writing Boolean expressions for circuits and constructing their truth tables are provided.
The document discusses number systems and conversions between different bases. It explains that computers use the binary system with bits representing 0s and 1s. 8 bits form a byte. Decimal, binary, octal and hexadecimal numbering systems are covered. Methods for converting between these bases are provided using division and remainders or grouping bits. Common powers and units used in computing like kilo, mega and giga are also defined. Exercises on converting values between the different number systems are included.
This document discusses different number systems including decimal, binary, octal, and hexadecimal. It explains how numbers are represented in each system using positional notation. Conversion between these number systems is demonstrated through examples. The document also covers signed integer representation methods like sign-and-magnitude, one's complement, and two's complement. Finally, it briefly introduces representation of characters using coding standards.
This document provides an overview of binary codes and Boolean algebra concepts. It discusses various binary codes like BCD, excess-3, and Gray codes. It also covers logical operations in Boolean algebra like AND, OR, and NOT. Properties of Boolean algebra like duality, De Morgan's laws, and absorption laws are explained. Digital logic gates are introduced as implementations of Boolean functions.
This document provides an overview of digital logic circuits and digital systems. It discusses binary logic, logic gates like NAND and NOR, Boolean algebra, decoders, adders, and the differences between analog and digital signals. It also covers representations of digital designs using truth tables, Boolean algebra, logic gate schematics, and logic simulations. Common logic gates, functions, identities, simplification techniques, and the duality principle of Boolean algebra are described.
This document outlines the topics covered in the 21EC201 - Digital Principles and System Design course. It includes an introduction to number systems, logic gates, combinational logic circuits, Boolean algebra, truth tables and Karnaugh maps. Specific topics mentioned are binary, decimal, octal and hexadecimal number systems, logic gates like AND, OR, NAND, NOR, XOR and XNOR, arithmetic operations in binary and conversions between different number systems.
This document discusses number representation systems used in computers, including binary, decimal, octal, and hexadecimal. It provides examples of converting between these different bases. Specifically, it covers:
1) Converting between decimal, binary, octal, and hexadecimal using positional notation.
2) Signed integer representation in binary, including sign-magnitude, one's complement, and two's complement. Examples are given of converting positive and negative decimals to these binary representations.
3) Storing integer, character, and floating point numbers in binary. Twos complement is described as the most common method for signed integer representation.
This document discusses number representation systems used in computers, including binary, decimal, octal, and hexadecimal. It provides examples of converting between these different bases. Specifically, it covers:
1) Converting between decimal, binary, octal, and hexadecimal using positional notation and place values.
2) Representing signed integers in binary using ones' complement and twos' complement notation.
3) Tables for converting binary numbers to octal and hexadecimal using place values of each base.
4) Examples of converting values between the different number bases both manually and using the provided conversion tables.
This document provides an introduction to number systems and binary logic gates. It discusses radix numbers including binary, octal, decimal, and hexadecimal. Methods for converting between bases are presented, along with examples. Binary arithmetic operations like addition, subtraction, and multiplication are also covered. The main logic gates - NOT, AND, OR, XOR, NAND, NOR, and XNOR - are defined through their truth tables. Circuit design is discussed, where combinations of gates can represent Boolean expressions. Examples of writing Boolean expressions for circuits and constructing their truth tables are provided.
The document discusses number systems and conversions between different bases. It explains that computers use the binary system with bits representing 0s and 1s. 8 bits form a byte. Decimal, binary, octal and hexadecimal numbering systems are covered. Methods for converting between these bases are provided using division and remainders or grouping bits. Common powers and units used in computing like kilo, mega and giga are also defined. Exercises on converting values between the different number systems are included.
This document discusses different number systems including decimal, binary, octal, and hexadecimal. It explains how numbers are represented in each system using positional notation. Conversion between these number systems is demonstrated through examples. The document also covers signed integer representation methods like sign-and-magnitude, one's complement, and two's complement. Finally, it briefly introduces representation of characters using coding standards.
This document provides an overview of binary codes and Boolean algebra concepts. It discusses various binary codes like BCD, excess-3, and Gray codes. It also covers logical operations in Boolean algebra like AND, OR, and NOT. Properties of Boolean algebra like duality, De Morgan's laws, and absorption laws are explained. Digital logic gates are introduced as implementations of Boolean functions.
This document provides an overview of digital logic circuits and digital systems. It discusses binary logic, logic gates like NAND and NOR, Boolean algebra, decoders, adders, and the differences between analog and digital signals. It also covers representations of digital designs using truth tables, Boolean algebra, logic gate schematics, and logic simulations. Common logic gates, functions, identities, simplification techniques, and the duality principle of Boolean algebra are described.
This document outlines the topics covered in the 21EC201 - Digital Principles and System Design course. It includes an introduction to number systems, logic gates, combinational logic circuits, Boolean algebra, truth tables and Karnaugh maps. Specific topics mentioned are binary, decimal, octal and hexadecimal number systems, logic gates like AND, OR, NAND, NOR, XOR and XNOR, arithmetic operations in binary and conversions between different number systems.
Number Systems - Arithmetic Operations - Binary Codes- Boolean Algebra and Logic Gates - Theorems and Properties of Boolean Algebra - Boolean Functions - Canonical and Standard Forms - Simplification of Boolean Functions using Karnaugh Map - Logic Gates – NAND and NOR Implementations.
The document discusses analogue and digital signals and number systems. It explains that the real world is analogue but digital signals are used for processing due to integrated circuits that can process digital data more easily. It then covers binary, octal, hexadecimal, and decimal number systems. Finally, it discusses representing negative numbers using sign-magnitude, 1's complement, and 2's complement representations and how arithmetic operations like addition and subtraction work using 2's complement.
This document summarizes different number systems used in computing including binary, octal, decimal, and hexadecimal. It explains how to convert between these number systems using theorems about their bases. Key topics covered include binary arithmetic, signed and unsigned integer representation, and how floating point numbers and characters are stored in binary format. Conversion charts are provided for binary to octal and hexadecimal. Representations of integers, characters, and floating point numbers in binary are also summarized.
This document provides an introduction and collection of problems related to bitwise operations and string manipulations. It begins with an introduction to bits, boolean functions, truth tables, and common bitwise operations like AND, OR, XOR, and shifts. The bulk of the document consists of 27 problems testing various concepts in bitwise logic and manipulations, such as simplifying boolean expressions, solving bitwise equations, and implementing logic gates using NAND gates. Solutions to the problems are not provided.
Digital devices like computers, watches, and phones use binary numbers encoded as signals with two values, 0 and 1. Basic logic gates like AND, OR, and NOT are used to build more complex digital circuits. Boolean algebra describes the logic operations performed by these circuits using rules for binary true/false values. Circuits add binary numbers by performing full adder logic on corresponding bits with sum and carry outputs.
This document provides an introduction to Boolean algebra and its applications in digital logic. It discusses how Boolean algebra was developed by George Boole in the 1800s as an algebra of logic to represent logical statements as either true or false. The document then explains how Boolean algebra is used to perform logical operations in digital computers by representing true as 1 and false as 0. It introduces the basic logical operators of AND, OR, and NOT and provides their truth tables. The rest of the document discusses topics such as logic gates, truth tables, minterms, maxterms, and how to realize Boolean functions using sum of products and product of sums forms.
Digital systems represent information using discrete binary values of 0 and 1 rather than continuous analog values. Binary numbers use a base-2 numbering system with place values that are powers of 2. There are various number systems like decimal, binary, octal and hexadecimal that use different number bases and represent the same number in different ways. Complements are used in binary arithmetic to perform subtraction by adding the 1's or 2's complement of a number. The 1's complement is obtained by inverting all bits, while the 2's complement is obtained by inverting all bits and adding 1.
Lec2 Intro to Computer Engineering by Hsien-Hsin Sean Lee Georgia Tech -- Num...Hsien-Hsin Sean Lee, Ph.D.
This document discusses number systems and binary arithmetic. It begins by explaining decimal and binary number representation, including place value and how to derive numbers in different bases. It then covers counting in binary, octal, and base-22 systems. Next, it discusses representing negative numbers using sign-magnitude, one's complement, and two's complement methods. Finally, it demonstrates binary addition and computation for both unsigned and signed numbers using two's complement.
Number System, Conversion, Decimal to Binary, Decimal to Octal, Decimal to Binary, Decimal to HexaDecimal, Binary to Decimal, Octal to Decimal, Hexadecimal to Decimal, Binary to Octal, Binary to Hexadecimal, Octal to Hexadecimal, BCD, Binary Addition
Numeral Systems: Positional and Non-Positional
Conversions between Positional Numeral Systems: Binary, Decimal and Hexadecimal
Representation of Numbers in Computer Memory
Exercises: Conversion between Different Numeral Systems
This document provides information about Boolean algebra. It begins with an introduction and table of contents. It then discusses the key concepts of Boolean algebra including constants, variables, functions, logical expressions, and logical operations. Features of Boolean algebra are presented, as well as the postulates and theorems. Laws of Boolean algebra like complement, AND, OR, commutative, associative, distributive, and absorption laws are defined. Examples are provided to illustrate concepts like consensus theorem, transposition theorem, De Morgan's theorem, and other theorems. The document also discusses binary coded decimal, excess-3 code, Gray code, and provides examples of arithmetic operations and conversions between different numeric systems.
The document discusses different number systems including binary, decimal, hexadecimal, and octal. It explains that number systems have a base, which is the number of unique digits used, and provides examples of how to convert between number systems. Binary coded decimal is also introduced as a way to efficiently store decimal numbers using a binary representation where each decimal digit is stored in 4 bits. Algorithms for binary addition and logic gates are briefly covered.
There are several number systems that can be used to represent numbers, which can be categorized as positional or non-positional. Commonly used positional systems include decimal, binary, octal, and hexadecimal. Different systems use different bases and symbols to represent values. Numbers can be converted between systems using techniques like successive division, weighted multiplication, or grouping bits. Understanding different number systems is important for both humans and computers.
This document provides lecture notes on digital system design. It covers topics like logic simplification, combinational logic design, understanding binary and other number systems, binary operations, and Boolean algebra. The first section discusses decimal, binary, octal and hexadecimal number systems. Later sections explain binary addition, subtraction, multiplication and conversions between number bases. Signed number representations like 1's complement and 2's complement are also introduced. Finally, the document discusses Boolean algebra, logic functions, truth tables, and basic logic gates like AND and INVERTER.
The document discusses digital and analog systems. It explains that digital systems represent information as discrete values using bits, whereas analog systems represent information as continuous values. It provides examples of digital and analog signals and discusses how a continuous analog signal can be converted to a discrete digital signal through sampling and quantization. It also covers binary, octal, and hexadecimal number systems and how to convert between them. Finally, it discusses binary addition and subtraction using complement representations.
This document discusses binary number representations. It begins with an overview of Boolean algebra and logical operations. It then covers representing positive integers using binary and other number systems such as hexadecimal and octal. Negative integers are represented using sign-magnitude, 1's complement, bias, and 2's complement representations. Properties of 2's complement include a unique representation for 0 and performing addition and subtraction. Floating point numbers and strings are also briefly mentioned.
Transistors can be used as switches in logic circuits to perform operations like AND and OR. AND logic requires both switches to be closed for current to flow, while OR logic allows current if either switch is closed. Binary addition is equivalent to an XOR operation plus an AND. Different number systems like binary, decimal, and hexadecimal can represent the same values using different bases.
The document discusses different number systems including binary, octal, decimal, and hexadecimal. It explains that number systems have a radix or base, which determines the set of symbols used and their positional values. The key representations for binary numbers discussed are sign-magnitude, one's complement, and two's complement, which provide different methods for representing positive and negative numbers. The document provides examples of addition, subtraction, multiplication, and division operations in binary.
The document discusses different number systems including binary, decimal, and hexadecimal. It defines base-N number systems and provides examples of decimal, binary, and hexadecimal numbers. Key concepts covered include positional notation, bits and bytes, addition and subtraction in different bases, signed numbers represented using sign-magnitude and two's complement, and arithmetic operations like addition and subtraction on signed binary numbers. Sign extension is introduced as an important concept for performing arithmetic on numbers of different bit-widths in a signed system.
The document discusses different number systems including binary, decimal, and hexadecimal. It defines base-N number systems and provides examples of decimal, binary, and hexadecimal numbers. Key concepts covered include positional notation, bits and bytes, addition and subtraction in different bases, signed numbers represented using sign-magnitude and two's complement, and arithmetic operations like addition and subtraction on signed binary numbers. Sign extension is introduced as an important concept for performing arithmetic on numbers of different bit-widths in a signed system.
Travis Hills' Endeavors in Minnesota: Fostering Environmental and Economic Pr...Travis Hills MN
Travis Hills of Minnesota developed a method to convert waste into high-value dry fertilizer, significantly enriching soil quality. By providing farmers with a valuable resource derived from waste, Travis Hills helps enhance farm profitability while promoting environmental stewardship. Travis Hills' sustainable practices lead to cost savings and increased revenue for farmers by improving resource efficiency and reducing waste.
The use of Nauplii and metanauplii artemia in aquaculture (brine shrimp).pptxMAGOTI ERNEST
Although Artemia has been known to man for centuries, its use as a food for the culture of larval organisms apparently began only in the 1930s, when several investigators found that it made an excellent food for newly hatched fish larvae (Litvinenko et al., 2023). As aquaculture developed in the 1960s and ‘70s, the use of Artemia also became more widespread, due both to its convenience and to its nutritional value for larval organisms (Arenas-Pardo et al., 2024). The fact that Artemia dormant cysts can be stored for long periods in cans, and then used as an off-the-shelf food requiring only 24 h of incubation makes them the most convenient, least labor-intensive, live food available for aquaculture (Sorgeloos & Roubach, 2021). The nutritional value of Artemia, especially for marine organisms, is not constant, but varies both geographically and temporally. During the last decade, however, both the causes of Artemia nutritional variability and methods to improve poorquality Artemia have been identified (Loufi et al., 2024).
Brine shrimp (Artemia spp.) are used in marine aquaculture worldwide. Annually, more than 2,000 metric tons of dry cysts are used for cultivation of fish, crustacean, and shellfish larva. Brine shrimp are important to aquaculture because newly hatched brine shrimp nauplii (larvae) provide a food source for many fish fry (Mozanzadeh et al., 2021). Culture and harvesting of brine shrimp eggs represents another aspect of the aquaculture industry. Nauplii and metanauplii of Artemia, commonly known as brine shrimp, play a crucial role in aquaculture due to their nutritional value and suitability as live feed for many aquatic species, particularly in larval stages (Sorgeloos & Roubach, 2021).
Number Systems - Arithmetic Operations - Binary Codes- Boolean Algebra and Logic Gates - Theorems and Properties of Boolean Algebra - Boolean Functions - Canonical and Standard Forms - Simplification of Boolean Functions using Karnaugh Map - Logic Gates – NAND and NOR Implementations.
The document discusses analogue and digital signals and number systems. It explains that the real world is analogue but digital signals are used for processing due to integrated circuits that can process digital data more easily. It then covers binary, octal, hexadecimal, and decimal number systems. Finally, it discusses representing negative numbers using sign-magnitude, 1's complement, and 2's complement representations and how arithmetic operations like addition and subtraction work using 2's complement.
This document summarizes different number systems used in computing including binary, octal, decimal, and hexadecimal. It explains how to convert between these number systems using theorems about their bases. Key topics covered include binary arithmetic, signed and unsigned integer representation, and how floating point numbers and characters are stored in binary format. Conversion charts are provided for binary to octal and hexadecimal. Representations of integers, characters, and floating point numbers in binary are also summarized.
This document provides an introduction and collection of problems related to bitwise operations and string manipulations. It begins with an introduction to bits, boolean functions, truth tables, and common bitwise operations like AND, OR, XOR, and shifts. The bulk of the document consists of 27 problems testing various concepts in bitwise logic and manipulations, such as simplifying boolean expressions, solving bitwise equations, and implementing logic gates using NAND gates. Solutions to the problems are not provided.
Digital devices like computers, watches, and phones use binary numbers encoded as signals with two values, 0 and 1. Basic logic gates like AND, OR, and NOT are used to build more complex digital circuits. Boolean algebra describes the logic operations performed by these circuits using rules for binary true/false values. Circuits add binary numbers by performing full adder logic on corresponding bits with sum and carry outputs.
This document provides an introduction to Boolean algebra and its applications in digital logic. It discusses how Boolean algebra was developed by George Boole in the 1800s as an algebra of logic to represent logical statements as either true or false. The document then explains how Boolean algebra is used to perform logical operations in digital computers by representing true as 1 and false as 0. It introduces the basic logical operators of AND, OR, and NOT and provides their truth tables. The rest of the document discusses topics such as logic gates, truth tables, minterms, maxterms, and how to realize Boolean functions using sum of products and product of sums forms.
Digital systems represent information using discrete binary values of 0 and 1 rather than continuous analog values. Binary numbers use a base-2 numbering system with place values that are powers of 2. There are various number systems like decimal, binary, octal and hexadecimal that use different number bases and represent the same number in different ways. Complements are used in binary arithmetic to perform subtraction by adding the 1's or 2's complement of a number. The 1's complement is obtained by inverting all bits, while the 2's complement is obtained by inverting all bits and adding 1.
Lec2 Intro to Computer Engineering by Hsien-Hsin Sean Lee Georgia Tech -- Num...Hsien-Hsin Sean Lee, Ph.D.
This document discusses number systems and binary arithmetic. It begins by explaining decimal and binary number representation, including place value and how to derive numbers in different bases. It then covers counting in binary, octal, and base-22 systems. Next, it discusses representing negative numbers using sign-magnitude, one's complement, and two's complement methods. Finally, it demonstrates binary addition and computation for both unsigned and signed numbers using two's complement.
Number System, Conversion, Decimal to Binary, Decimal to Octal, Decimal to Binary, Decimal to HexaDecimal, Binary to Decimal, Octal to Decimal, Hexadecimal to Decimal, Binary to Octal, Binary to Hexadecimal, Octal to Hexadecimal, BCD, Binary Addition
Numeral Systems: Positional and Non-Positional
Conversions between Positional Numeral Systems: Binary, Decimal and Hexadecimal
Representation of Numbers in Computer Memory
Exercises: Conversion between Different Numeral Systems
This document provides information about Boolean algebra. It begins with an introduction and table of contents. It then discusses the key concepts of Boolean algebra including constants, variables, functions, logical expressions, and logical operations. Features of Boolean algebra are presented, as well as the postulates and theorems. Laws of Boolean algebra like complement, AND, OR, commutative, associative, distributive, and absorption laws are defined. Examples are provided to illustrate concepts like consensus theorem, transposition theorem, De Morgan's theorem, and other theorems. The document also discusses binary coded decimal, excess-3 code, Gray code, and provides examples of arithmetic operations and conversions between different numeric systems.
The document discusses different number systems including binary, decimal, hexadecimal, and octal. It explains that number systems have a base, which is the number of unique digits used, and provides examples of how to convert between number systems. Binary coded decimal is also introduced as a way to efficiently store decimal numbers using a binary representation where each decimal digit is stored in 4 bits. Algorithms for binary addition and logic gates are briefly covered.
There are several number systems that can be used to represent numbers, which can be categorized as positional or non-positional. Commonly used positional systems include decimal, binary, octal, and hexadecimal. Different systems use different bases and symbols to represent values. Numbers can be converted between systems using techniques like successive division, weighted multiplication, or grouping bits. Understanding different number systems is important for both humans and computers.
This document provides lecture notes on digital system design. It covers topics like logic simplification, combinational logic design, understanding binary and other number systems, binary operations, and Boolean algebra. The first section discusses decimal, binary, octal and hexadecimal number systems. Later sections explain binary addition, subtraction, multiplication and conversions between number bases. Signed number representations like 1's complement and 2's complement are also introduced. Finally, the document discusses Boolean algebra, logic functions, truth tables, and basic logic gates like AND and INVERTER.
The document discusses digital and analog systems. It explains that digital systems represent information as discrete values using bits, whereas analog systems represent information as continuous values. It provides examples of digital and analog signals and discusses how a continuous analog signal can be converted to a discrete digital signal through sampling and quantization. It also covers binary, octal, and hexadecimal number systems and how to convert between them. Finally, it discusses binary addition and subtraction using complement representations.
This document discusses binary number representations. It begins with an overview of Boolean algebra and logical operations. It then covers representing positive integers using binary and other number systems such as hexadecimal and octal. Negative integers are represented using sign-magnitude, 1's complement, bias, and 2's complement representations. Properties of 2's complement include a unique representation for 0 and performing addition and subtraction. Floating point numbers and strings are also briefly mentioned.
Transistors can be used as switches in logic circuits to perform operations like AND and OR. AND logic requires both switches to be closed for current to flow, while OR logic allows current if either switch is closed. Binary addition is equivalent to an XOR operation plus an AND. Different number systems like binary, decimal, and hexadecimal can represent the same values using different bases.
The document discusses different number systems including binary, octal, decimal, and hexadecimal. It explains that number systems have a radix or base, which determines the set of symbols used and their positional values. The key representations for binary numbers discussed are sign-magnitude, one's complement, and two's complement, which provide different methods for representing positive and negative numbers. The document provides examples of addition, subtraction, multiplication, and division operations in binary.
The document discusses different number systems including binary, decimal, and hexadecimal. It defines base-N number systems and provides examples of decimal, binary, and hexadecimal numbers. Key concepts covered include positional notation, bits and bytes, addition and subtraction in different bases, signed numbers represented using sign-magnitude and two's complement, and arithmetic operations like addition and subtraction on signed binary numbers. Sign extension is introduced as an important concept for performing arithmetic on numbers of different bit-widths in a signed system.
The document discusses different number systems including binary, decimal, and hexadecimal. It defines base-N number systems and provides examples of decimal, binary, and hexadecimal numbers. Key concepts covered include positional notation, bits and bytes, addition and subtraction in different bases, signed numbers represented using sign-magnitude and two's complement, and arithmetic operations like addition and subtraction on signed binary numbers. Sign extension is introduced as an important concept for performing arithmetic on numbers of different bit-widths in a signed system.
Travis Hills' Endeavors in Minnesota: Fostering Environmental and Economic Pr...Travis Hills MN
Travis Hills of Minnesota developed a method to convert waste into high-value dry fertilizer, significantly enriching soil quality. By providing farmers with a valuable resource derived from waste, Travis Hills helps enhance farm profitability while promoting environmental stewardship. Travis Hills' sustainable practices lead to cost savings and increased revenue for farmers by improving resource efficiency and reducing waste.
The use of Nauplii and metanauplii artemia in aquaculture (brine shrimp).pptxMAGOTI ERNEST
Although Artemia has been known to man for centuries, its use as a food for the culture of larval organisms apparently began only in the 1930s, when several investigators found that it made an excellent food for newly hatched fish larvae (Litvinenko et al., 2023). As aquaculture developed in the 1960s and ‘70s, the use of Artemia also became more widespread, due both to its convenience and to its nutritional value for larval organisms (Arenas-Pardo et al., 2024). The fact that Artemia dormant cysts can be stored for long periods in cans, and then used as an off-the-shelf food requiring only 24 h of incubation makes them the most convenient, least labor-intensive, live food available for aquaculture (Sorgeloos & Roubach, 2021). The nutritional value of Artemia, especially for marine organisms, is not constant, but varies both geographically and temporally. During the last decade, however, both the causes of Artemia nutritional variability and methods to improve poorquality Artemia have been identified (Loufi et al., 2024).
Brine shrimp (Artemia spp.) are used in marine aquaculture worldwide. Annually, more than 2,000 metric tons of dry cysts are used for cultivation of fish, crustacean, and shellfish larva. Brine shrimp are important to aquaculture because newly hatched brine shrimp nauplii (larvae) provide a food source for many fish fry (Mozanzadeh et al., 2021). Culture and harvesting of brine shrimp eggs represents another aspect of the aquaculture industry. Nauplii and metanauplii of Artemia, commonly known as brine shrimp, play a crucial role in aquaculture due to their nutritional value and suitability as live feed for many aquatic species, particularly in larval stages (Sorgeloos & Roubach, 2021).
Phenomics assisted breeding in crop improvementIshaGoswami9
As the population is increasing and will reach about 9 billion upto 2050. Also due to climate change, it is difficult to meet the food requirement of such a large population. Facing the challenges presented by resource shortages, climate
change, and increasing global population, crop yield and quality need to be improved in a sustainable way over the coming decades. Genetic improvement by breeding is the best way to increase crop productivity. With the rapid progression of functional
genomics, an increasing number of crop genomes have been sequenced and dozens of genes influencing key agronomic traits have been identified. However, current genome sequence information has not been adequately exploited for understanding
the complex characteristics of multiple gene, owing to a lack of crop phenotypic data. Efficient, automatic, and accurate technologies and platforms that can capture phenotypic data that can
be linked to genomics information for crop improvement at all growth stages have become as important as genotyping. Thus,
high-throughput phenotyping has become the major bottleneck restricting crop breeding. Plant phenomics has been defined as the high-throughput, accurate acquisition and analysis of multi-dimensional phenotypes
during crop growing stages at the organism level, including the cell, tissue, organ, individual plant, plot, and field levels. With the rapid development of novel sensors, imaging technology,
and analysis methods, numerous infrastructure platforms have been developed for phenotyping.
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
The ability to recreate computational results with minimal effort and actionable metrics provides a solid foundation for scientific research and software development. When people can replicate an analysis at the touch of a button using open-source software, open data, and methods to assess and compare proposals, it significantly eases verification of results, engagement with a diverse range of contributors, and progress. However, we have yet to fully achieve this; there are still many sociotechnical frictions.
Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
Our observation is that multiple layers — hardware, operating systems, third-party libraries, software versions, input data, compile-time options, and parameters — are subject to variability that exacerbates frictions but is also essential for achieving robust, generalizable results and fostering innovation. I will first review the literature, providing evidence of how the complex variability interactions across these layers affect qualitative and quantitative software properties, thereby complicating the reproduction and replication of scientific studies in various fields.
I will then present some software engineering and AI techniques that can support the strategic exploration of variability spaces. These include the use of abstractions and models (e.g., feature models), sampling strategies (e.g., uniform, random), cost-effective measurements (e.g., incremental build of software configurations), and dimensionality reduction methods (e.g., transfer learning, feature selection, software debloating).
I will finally argue that deep variability is both the problem and solution of frictionless reproducibility, calling the software science community to develop new methods and tools to manage variability and foster reproducibility in software systems.
Exposé invité Journées Nationales du GDR GPL 2024
Unlocking the mysteries of reproduction: Exploring fecundity and gonadosomati...AbdullaAlAsif1
The pygmy halfbeak Dermogenys colletei, is known for its viviparous nature, this presents an intriguing case of relatively low fecundity, raising questions about potential compensatory reproductive strategies employed by this species. Our study delves into the examination of fecundity and the Gonadosomatic Index (GSI) in the Pygmy Halfbeak, D. colletei (Meisner, 2001), an intriguing viviparous fish indigenous to Sarawak, Borneo. We hypothesize that the Pygmy halfbeak, D. colletei, may exhibit unique reproductive adaptations to offset its low fecundity, thus enhancing its survival and fitness. To address this, we conducted a comprehensive study utilizing 28 mature female specimens of D. colletei, carefully measuring fecundity and GSI to shed light on the reproductive adaptations of this species. Our findings reveal that D. colletei indeed exhibits low fecundity, with a mean of 16.76 ± 2.01, and a mean GSI of 12.83 ± 1.27, providing crucial insights into the reproductive mechanisms at play in this species. These results underscore the existence of unique reproductive strategies in D. colletei, enabling its adaptation and persistence in Borneo's diverse aquatic ecosystems, and call for further ecological research to elucidate these mechanisms. This study lends to a better understanding of viviparous fish in Borneo and contributes to the broader field of aquatic ecology, enhancing our knowledge of species adaptations to unique ecological challenges.
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...Sérgio Sacani
Context. With a mass exceeding several 104 M⊙ and a rich and dense population of massive stars, supermassive young star clusters
represent the most massive star-forming environment that is dominated by the feedback from massive stars and gravitational interactions
among stars.
Aims. In this paper we present the Extended Westerlund 1 and 2 Open Clusters Survey (EWOCS) project, which aims to investigate
the influence of the starburst environment on the formation of stars and planets, and on the evolution of both low and high mass stars.
The primary targets of this project are Westerlund 1 and 2, the closest supermassive star clusters to the Sun.
Methods. The project is based primarily on recent observations conducted with the Chandra and JWST observatories. Specifically,
the Chandra survey of Westerlund 1 consists of 36 new ACIS-I observations, nearly co-pointed, for a total exposure time of 1 Msec.
Additionally, we included 8 archival Chandra/ACIS-S observations. This paper presents the resulting catalog of X-ray sources within
and around Westerlund 1. Sources were detected by combining various existing methods, and photon extraction and source validation
were carried out using the ACIS-Extract software.
Results. The EWOCS X-ray catalog comprises 5963 validated sources out of the 9420 initially provided to ACIS-Extract, reaching a
photon flux threshold of approximately 2 × 10−8 photons cm−2
s
−1
. The X-ray sources exhibit a highly concentrated spatial distribution,
with 1075 sources located within the central 1 arcmin. We have successfully detected X-ray emissions from 126 out of the 166 known
massive stars of the cluster, and we have collected over 71 000 photons from the magnetar CXO J164710.20-455217.
hematic appreciation test is a psychological assessment tool used to measure an individual's appreciation and understanding of specific themes or topics. This test helps to evaluate an individual's ability to connect different ideas and concepts within a given theme, as well as their overall comprehension and interpretation skills. The results of the test can provide valuable insights into an individual's cognitive abilities, creativity, and critical thinking skills
The binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
BREEDING METHODS FOR DISEASE RESISTANCE.pptxRASHMI M G
Plant breeding for disease resistance is a strategy to reduce crop losses caused by disease. Plants have an innate immune system that allows them to recognize pathogens and provide resistance. However, breeding for long-lasting resistance often involves combining multiple resistance genes
This presentation explores a brief idea about the structural and functional attributes of nucleotides, the structure and function of genetic materials along with the impact of UV rays and pH upon them.
When I was asked to give a companion lecture in support of ‘The Philosophy of Science’ (https://shorturl.at/4pUXz) I decided not to walk through the detail of the many methodologies in order of use. Instead, I chose to employ a long standing, and ongoing, scientific development as an exemplar. And so, I chose the ever evolving story of Thermodynamics as a scientific investigation at its best.
Conducted over a period of >200 years, Thermodynamics R&D, and application, benefitted from the highest levels of professionalism, collaboration, and technical thoroughness. New layers of application, methodology, and practice were made possible by the progressive advance of technology. In turn, this has seen measurement and modelling accuracy continually improved at a micro and macro level.
Perhaps most importantly, Thermodynamics rapidly became a primary tool in the advance of applied science/engineering/technology, spanning micro-tech, to aerospace and cosmology. I can think of no better a story to illustrate the breadth of scientific methodologies and applications at their best.
Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luísa Pinho
Functional Magnetic Resonance Imaging (fMRI) provides means to characterize brain activations in response to behavior. However, cognitive neuroscience has been limited to group-level effects referring to the performance of specific tasks. To obtain the functional profile of elementary cognitive mechanisms, the combination of brain responses to many tasks is required. Yet, to date, both structural atlases and parcellation-based activations do not fully account for cognitive function and still present several limitations. Further, they do not adapt overall to individual characteristics. In this talk, I will give an account of deep-behavioral phenotyping strategies, namely data-driven methods in large task-fMRI datasets, to optimize functional brain-data collection and improve inference of effects-of-interest related to mental processes. Key to this approach is the employment of fast multi-functional paradigms rich on features that can be well parametrized and, consequently, facilitate the creation of psycho-physiological constructs to be modelled with imaging data. Particular emphasis will be given to music stimuli when studying high-order cognitive mechanisms, due to their ecological nature and quality to enable complex behavior compounded by discrete entities. I will also discuss how deep-behavioral phenotyping and individualized models applied to neuroimaging data can better account for the subject-specific organization of domain-general cognitive systems in the human brain. Finally, the accumulation of functional brain signatures brings the possibility to clarify relationships among tasks and create a univocal link between brain systems and mental functions through: (1) the development of ontologies proposing an organization of cognitive processes; and (2) brain-network taxonomies describing functional specialization. To this end, tools to improve commensurability in cognitive science are necessary, such as public repositories, ontology-based platforms and automated meta-analysis tools. I will thus discuss some brain-atlasing resources currently under development, and their applicability in cognitive as well as clinical neuroscience.
3. Understanding Decimal Numbers
• Decimal numbers are made of decimal
digits: (0,1,2,3,4,5,6,7,8,9)
• But how many items does a decimal
number represent?
8653 = 8x103 + 6x102 + 5x101 + 3x100
• What about fractions?
97654.35 = 9x104 + 7x103 + 6x102 + 5x101
+ 4x100 + 3x10-1 + 5x10-2
In formal notation -> (97654.35)10
• Why do we use 10 digits, anyway?
4. Understanding Octal Numbers
• Octal numbers are made of octal digits:
(0,1,2,3,4,5,6,7)
• How many items does an octal number
represent?
(4536)8 = 4x83 + 5x82 + 3x81 + 6x80 = (1362)10
• What about fractions?
(465.27)8 = 4x82 + 6x81 + 5x80 + 2x8-1 + 7x8-2
• Octal numbers don’t use digits 8 or 9
• Who would use octal number, anyway?
5. Understanding Binary Numbers
• Binary numbers are made of binary digits
(bits):
0 and 1
• How many items does an binary number
represent?
(1011)2 = 1x23 + 0x22 + 1x21 + 1x20 = (11)10
• What about fractions?
(110.10)2 = 1x22 + 1x21 + 0x20 + 1x2-1 + 0x2-2
• Groups of eight bits are called a byte
(11001001) 2
• Groups of four bits are called a nibble.
(1101) 2
6. Understanding Hexadecimal Numbers
• Hexadecimal numbers are made of 16 digits:
(0,1,2,3,4,5,6,7,8,9,A, B, C, D, E, F)
• How many items does an hex number represent?
(3A9F)16 = 3x163 + 10x162 + 9x161 + 15x160 = 1499910
• What about fractions?
(2D3.5)16 = 2x162 + 13x161 + 3x160 + 5x16-1 =
723.312510
• Note that each hexadecimal digit can be represented
with four bits.
(1110) 2 = (E)16
• Groups of four bits are called a nibble.
(1110) 2
7.
8.
9. Converting Binary to Decimal
• To Convert to decimal, use decimal
arithmetic to sum the weighted
powers of two:
• Converting 110102 to N10:
N10 = 1 x 24 x 1x 23 + 0 x 22 + 21 + 0 + 20
= 26
10.
11.
12.
13.
14. Converting Between Base 16 and Base 2
• Conversion is easy!
• Determine 4-bit value for each hex digit
• Note that there are 24 = 16 different values of
four bits
• Easier to read and write in hexadecimal.
• Representations are equivalent!
3A9F16 = 0011 1010 1001 11112
3 A 9 F
15. Converting Between Base 16 and Base 8
1. Convert from Base 16 to Base 2
2. Regroup bits into groups of three starting from
right
3. Ignore leading zeros
4. Each group of three bits forms an octal digit.
3A9F16 = 0011 1010 1001 11112
3 A 9 F
352378 = 011 101 010 011 1112
5 2 3 7
3
16. 16
Conversion of Bases
Example: Base 8 to base 10
(432.2)8 = 4 82 + 3 81 + 2 80 + 2 8-1 = (282.25)10
Example: Base 2 to base 10
(1101.01)2 = 1 23 + 1 22 + 0 21 + 1 20 + 0 2-1 + 1 2-2 = (13.25)10
Base b1 to b2, where b1 > b2:
17. 17
Conversion of Bases (Contd.)
Example: Convert (548)10 to base 8
Thus, (548)10 = (1044)8
Thus, (345)10 = (1333)6
Example: Convert (345)10 to base 6
21. Binary and Weighted Codes
• Although binary systems have advantages in digital computers
(to control the switches), humans work in decimal systems.
• It is convenient to represent decimal digits by sequence of
binary digits.
• Several coding techniques have been developed to do so
Decimal digits: 0, 1, …, 9 (10) can be represented by 4 bits.
• Since, we need 10 out of 16 values, several codes possible.
• Weighted Codes: If x1, x2, x3, x4 are the binary digits, with
weights w1, w2, w3, w4, then the decimal digit is:
N=w4x4+w3x3+w2x2+w1x1
We say, the sequence (x1, x2, x3, x4) denotes the code word for
N.
21
22. 22
Binary Codes
BCD
Self-complementing code: Code word of 9’s complement of N obtained
by interchanging 1’s and 0’s in the code word of N
Self-complementing Codes
Is this
unique?
23. 23
Nonweighted Codes
Add 3 to
BCD
Successive code words
differ in only one digit
Can you see some
interesting
properties in the
excess-3 code?
27. 27
Hamming Codes: Single Error-correcting
Minimum distance for SEC or double-error detecting (DED) codes = 3
Example: {000,111}
Minimum distance for SEC and DED codes = 4
No. of information bits = m
No. of parity check bits, p1, p2, …, pk = k
No. of bits in the code word = m+k
Assign a decimal value to each of the m+k bits: from 1 to MSB to m+k to
LSB
Perform k parity checks on selected bits of each code word: record results
as 0 or 1
• Form a binary number (called position number), c1c2…ck, with the k
parity checks
28. 28
Hamming Codes (Contd.)
No. of parity check bits, k, must satisfy: 2k >= m+k+1
Example: if m = 4 then k =3
Place check bits at the following locations: 1, 2, 4, …, 2k-1
Example code word: 1100110
• Check bits: p1= 1, p2 = 1, p3 = 0
• Information bits: 0, 1, 1, 0
29. 29
Hamming Code Construction
Select p1 to establish even parity in positions: 1, 3, 5, 7
Select p2 to establish even parity in positions: 2, 3, 6, 7
Select p3 to establish even parity in positions: 4, 5, 6, 7
31. 31
Hamming Code for BCD
Position: 1 2 3 4 5 6 7
Intended message: 1 1 0 1 0 0 1
Message received: 1 1 0 1 1 0 1
4-5-6-7 parity check: 1 1 0 1 c1 = 1 since parity is odd
2-3-6-7 parity check: 1 0 0 1 c2 = 0 since parity is even
1-3-5-7 parity check: 1 0 1 1 c3 = 1 since parity is odd
32. 32
Boolean Algebra
• Boolean Algebra named after George Boole who
used it to study human logical reasoning – calculus
of proposition.
• Elements : true or false ( 0, 1)
• Operations: a OR b; a AND b, NOT a
e.g. 0 OR 1 = 1 0 OR 0 = 0
1 AND 1 = 1 1 AND 0 = 0
NOT 0 = 1 NOT 1 = 0
What is an Algebra? (e.g. algebra of integers)
set of elements (e.g. 0,1,2,..)
set of operations (e.g. +, -, *,..)
postulates/axioms (e.g. 0+x=x,..)
33. Boolean function
• Boolean function: Mapping from Boolean
variables to a Boolean value.
• Boolean algebra: Deals with binary variables and
logic operations operating on those variables.
34. Basic Identities of Boolean Algebra
(Existence of 1 and 0 element)
(1) x + 0 = x
(2) x · 0 = 0
(3) x + 1 = 1
(4) x · 1 = 1
35. Basic Identities of Boolean Algebra
(Existence of complement)
(5) x + x = x
(6) x · x = x
(7) x + x’ = x
(8) x · x’ = 0
36. Basic Identities of Boolean Algebra
Commutativity:
(9) x + y = y + x
(10) xy = yx
Associativity:
(11) x + ( y + z ) = ( x + y ) + z
(12) x (yz) = (xy) z
Distributivity:
(13) x ( y + z ) = xy + xz
(14) x + yz = ( x + y )( x + z)
37. Basic Identities of Boolean
Algebra
De-Morgan’s Theorem:
(15) ( x + y )’ = x’ y’
(16) ( xy )’ = x’ + y’
Generalized DeMorgan's Theorem
(a) (a + b + … z)' = a'b' … z'
(b) (a.b … z)' = a' + b' + … z‘
Involution:
(17) (x’)’ = x
38.
39. Function Minimization using Boolean Algebra
Examples:
(a) a + ab = a(1+b)=a
(b) a(a + b) = a.a +ab=a+ab=a(1+b)=a.
(c) a + a'b = (a + a')(a + b)=1(a + b) =a+b
(d) a(a' + b) = a. a' +ab=0+ab=ab
Show that;
1- ab + ab' = a
2- (a + b)(a + b') = a
1- ab + ab' = a(b+b') = a.1=a
2- (a + b)(a + b') = a.a +a.b' +a.b+b.b'
= a + a.b' +a.b + 0
= a + a.(b' +b) + 0
= a + a.1 + 0
= a + a = a
40. More De-Morgan's example
Show that: (a(b + z(x + a')))' =a' + b' (z' + x')
(a(b + z(x + a')))' = a' + (b + z(x + a'))'
= a' + b' (z(x + a'))'
= a' + b' (z' + (x + a')')
= a' + b' (z' + x'(a')')
= a' + b' (z' + x'a)
=a‘+b' z' + b'x'a
=(a‘+ b'x'a) + b' z'
=(a‘+ b'x‘)(a +a‘) + b' z'
= a‘+ b'x‘+ b' z‘
= a' + b' (z' + x')
42. AND Function
Output Y is TRUE if inputs A AND B 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
43. OR Function
Output Y is TRUE if input A OR B is TRUE, else it
is FALSE.
Logic Symbol
Text Description
Truth Table
Boolean Expression Y = A + B
OR Symbol
A
B
Y
OR
INPUTS OUTPUT
A B Y
0 0 0
0 1 1
1 0 1
1 1 1
OR Gate Truth Table
44. 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 Y
NOT
NOT
Bar
Y = A
Y = A’
Alternative Notation
Y = !A
45. NAND Function
Output Y is FALSE if inputs A AND B are TRUE,
else it is TRUE.
Logic Symbol
Text Description
Truth Table
Boolean Expression
A
B
Y
NAND
A bubble is an inverter
This is an AND Gate with an inverted output
Y = A x B = AB
INPUTS OUTPUT
A B Y
0 0 1
0 1 1
1 0 1
1 1 0
NAND Gate Truth Table
46. NOR Function
Output Y is FALSE if input A OR B is TRUE, else it
is TRUE.
Logic Symbol
Text Description
Truth Table
Boolean Expression Y = A + B
A
B
Y
NOR
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
47. SOP Given a Table of Combinations
– What is the SOP form for the following 3 input / 1
output digital device?
S A B f
0 0 0 0
0 0 1 0
0 1 0 1
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 0
1 1 1 1
48. • Computing the SOP (2)
– This SOP has 4 minterms:
• f = S'AB' + S'AB + SA'B + SAB
S A B f minterm name
0 1 0 1 m2
0 1 1 1 m3
1 0 1 1 m5
1 1 1 1 m7
49. • Canonical SOP
– Boolean functions can use shorthand notation when
in SOP form:
• f = S'AB' + S'AB + SA'B + SAB
f(S,A,B) = (m2,m3,m5,m7)
or
f(S,A,B) = m(2,3,5,7)
51. • Product of Sums Form
– An alternate canonical “two-level” format
• “Product of sums” POS
• Two levels
– OR level followed by AND level
– Again, NOT doesn’t count as a level
• Not a common as SOP, but can be useful in some situations
– Which ones?
52. • Computing the POS
– Identify rows with “0” on output (f = 0)
– Represent the input for each 0 row as a maxterm
• A logical “sum” of the input bits which guarantees that term
will be “0” (sum of literals)
A B f
0 0 0
0 1 1
1 0 0
1 1 0
54. Terminology/Definition
• Literal
– A variable or its complement
• Logically adjacent terms
– Two minterms are logically adjacent if
they differ in only one variable position
– Ex:
abc abc
and
m6 and m2 are logically adjacent
Note:
abc abc a a bc bc
Or, logically adjacent terms can be combined
55. Terminology/Definition
• Implicant
– Product term that could be used to cover minterms of
a function
• Prime Implicant
– An implicant that is not part of another implicant
• Essential Prime Implicant
– An implicant that covers at least one minterm that is
not contained in another prime implicant
• Cover
– A minterm that has been used in at least one group
56. Guidelines for Simplifying Functions
• Each square on a K-map of n
variables has n logically adjacent
squares. (i.e. differing in exactly one
variable)
• When combing squares, always
group in powers of 2m , where
m=0,1,2,….
• In general, grouping 2m variables
eliminates m variables.
57. Guidelines for Simplifying Functions
• Group as many squares as possible.
This eliminates the most variables.
• Make as few groups as possible.
Each group represents a separate
product term.
• You must cover each minterm at
least once. However, it may be
covered more than once.
58. K-map Simplification
Procedure
• Plot the K-map
• Circle all prime implicants on the K-
map
• Identify and select all essential prime
implicants for the cover.
• Select a minimum subset of the
remaining prime implicants to
complete the cover.
• Read the K-map
59. Example
• Use a K-Map to simplify the following
Boolean expression
, , 1, 2,3,5,6
F a b c m
62. Three-Variable K-Map
Example
ab
c 00 01 11 10
0
1
Step 3: Identify Essential Prime Implicants
1 1 1
1
, , 1, 2,3,5,6
F a b c m
1
EPI
EPI
PI
PI
63. Three-Variable K-Map
Example
ab
c 00 01 11 10
0
1
Step 4: Select minimum subset of remaining
Prime Implicants to complete the cover.
1 1 1
1
, , 1, 2,3,5,6
F a b c m
1
EPI
PI
EPI
68. Three-Variable K-Map
Example
ab
c 00 01 11 10
0
1
Step 2: Circle Prime Implicants
1
1
1
1
, , 2,3,6,7
F a b c m
Wrong!!
We really
should draw
A circle around
all four 1’s
69. Three-Variable K-Map
Example
ab
c 00 01 11 10
0
1
Step 3: Identify Essential Prime Implicants
EPI
EPI
, , 2,3,6,7
F a b c m
1
1
1
1
Wrong!!
We really
should draw
A circle around
all four 1’s
70. Three-Variable K-Map
Example
ab
c 00 01 11 10
0
1
Step 4: Select Remaining Prime Implicants to
complete the cover.
EPI
EPI
1
1
1
1
, , 2,3,6,7
F a b c m
84. Example
• Use a K-Map to simplify the following
Boolean expression
, , , 0,2,6,8,12,13,15
3,9,10
F a b c d m
d
D=Don’t care (i.e. either 1 or 0)
85. Four-variable K-Map
ab
cd 00 01 11 10
00
01
11
10
1
1
d
1
1
1
1
1
, , , 0, 2,6,8,12,13,15 3, 4,9
F a b c d m d
d
d
87. KARNAUGH MAP
Karnaugh Map for an n-input digital logic circuit (n-variable sum-of-products
form of Boolean Function, or Truth Table) is
- Rectangle divided into 2n cells
- Each cell is associated with a Minterm
- An output(function) value for each input value associated with a
mintern is written in the cell representing the minterm
→ 1-cell, 0-cell
Each Minterm is identified by a decimal number whose binary representation
is identical to the binary interpretation of the input values of the minterm.
x F
0 1
1 0
x
0
1
0
1
x
0
1
0
1
Karnaugh Map
value
of F
Identification
of the cell
x y F
0 0 0
0 1 1
1 0 1
1 1 1
y
x 0 1
0
1
0 1
2 3
y
x 0 1
0
1
0 1
1 0
F(x) =
F(x,y) = (1,2)
1-cell
(1)
Map Simplification
89. MAP SIMPLIFICATION - 2
ADJACENT CELLS -
Adjacent cells
- binary identifications are different in one bit
→ minterms associated with the adjacent
cells have one variable complemented each other
Cells (1,0) and (1,1) are adjacent
Minterms for (1,0) and (1,1) are
x • y’ --> x=1, y=0
x • y --> x=1, y=1
F = xy’+ xy can be reduced to F = x
From the map
Rule: xy’ +xy = x(y+y’) = x
x
y
0 1
0
1 1 1
0 0
(2,3)
F(x,y) =
2 adjacent cells xy’ and xy
→ merge them to a larger cell x
= xy’+ xy
= x
Map Simplification
90. MAP SIMPLIFICATION - MORE
THAN 2 CELLS -
u’v’w’x’ + u’v’w’x + u’v’wx + u’v’wx’
= u’v’w’(x’+x) + u’v’w(x+x’)
= u’v’w’ + u’v’w
= u’v’(w’+w)
= u’v’
uv
wx
1 1 1 1
1 1
1 1
uv
wx
1 1 1 1
1 1
1 1
u
v
w
x
u
v
w
x
u’v’
uw
u’x’
v’x
1 1
1 1
vw’
u’v’w’x’+u’v’w’x+u’vw’x’+u’vw’x+uvw’x’+uvw’x+uv’w’x’+uv’w’x
= u’v’w’(x’+x) + u’vw’(x’+x) + uvw’(x’+x) + uv’w’(x’+x)
= u’(v’+v)w’ + u(v’+v)w’
= (u’+u)w’ = w’
Map Simplification
u
v
w
x
uv
wx
1 1
1 1
1 1
1 1
u
v
uv
1 1
1 1
1 1
1 1
1 1 1 1
x
w’
u
V’
w
91. MAP SIMPLIFICATION
(0,1), (0,2), (0,4), (0,8)
Adjacent Cells of 1
Adjacent Cells of 0
(1,0), (1,3), (1,5), (1,9)
...
...
Adjacent Cells of 15
(15,7), (15,11), (15,13), (15,14)
uv
wx
00 01 11 10
00
01 0 0 0 0
11 0 1 1 0
10 0 1 0 0
1 1 0 1
F(u,v,w,x) = (0,1,2,9,13,15)
u
v
w
x
Merge (0,1) and (0,2)
--> u’v’w’ + u’v’x’
Merge (1,9)
--> v’w’x
Merge (9,13)
--> uw’x
Merge (13,15)
--> uvx
F = u’v’w’ + u’v’x’ + v’w’x + uw’x + uvx
But (9,13) is covered by (1,9) and (13,15)
F = u’v’w’ + u’v’x’ + v’w’x + uvx
Map Simplification
0 0 0 0
1 1 0 1
0 1 1 0
0 1 0 0
92. IMPLEMENTATION OF K-MAPS - Sum-of-Products Form -
Logic function represented by a Karnaugh map
can be implemented in the form of I-AND-OR
A cell or a collection of the adjacent 1-cells can
be realized by an AND gate, with some inversion of the input variables.
x
y
z
x’
y’
z’
x’
y
z’
x
y
z’
1 1
1
F(x,y,z) = (0,2,6)
1 1
1
x’
z’
y
z’
Map Simplification
x’
y
x
y
z’
x’
y’
z’
F
x
z
y
z
F
I AND OR
z’
93. IMPLEMENTATION OF K-MAPS - Product-of-Sums Form -
Logic function represented by a Karnaugh map
can be implemented in the form of I-OR-AND
If we implement a Karnaugh map using 0-cells,
the complement of F, i.e., F’, can be obtained.
Thus, by complementing F’ using DeMorgan’s
theorem F can be obtained
F(x,y,z) = (0,2,6)
x
y
z
x
y’
z
F’ = xy’ + z
F = (xy’)z’
= (x’ + y)z’
x
y
z
F
I OR AND
Map Simplification
0 0
1 1
0 0 0 1
94. IMPLEMENTATION OF K-MAPS
- Don’t-Care Conditions -
In some logic circuits, the output responses
for some input conditions are don’t care
whether they are 1 or 0.
In K-maps, don’t-care conditions are represented
by d’s in the corresponding cells.
Don’t-care conditions are useful in minimizing
the logic functions using K-map.
- Can be considered either 1 or 0
- Thus increases the chances of merging cells into the larger cells
--> Reduce the number of variables in the product terms
x
y
z
1 d d 1
d 1
x’
yz’
x
y
z
F
Map Simplification
96. Adding Two 1-bit Numbers
Let us add two 1 bit numbers : a and b
0 + 0 = 00
1 + 0 = 01
0 + 1 = 01
1 + 1 = 10
The lsb of the result is known, as the sum,
and the msb is known as the carry
103. Circuit for the Full Adder
a
b
a
b
Full
adder
a
b
S
a
b
cin
cin
cout
cin
s
cin
c out
104. Lan-Da Van DCD-
1-Bit Full Adder
Full-Adder
The arithmetic sum of three
input bits
three input bits
x, y: two significant bits
z: the carry bit from the
previous lower significant bit
Two output bits: C, S
Sum Carry
106. Logic Diagram of 1-Bit
Full Adder
S = x'y'z+x'yz'+ xy'z'+xyz
= x’(yz) +x(yz)’ = xyz
C = xy + xz + yz
= xy + xyz + xy’z + xyz + x’yz
= xy + z (xy + xy)
= xy + z (xy)
107. Addition of two n bit numbers
We start from the lsb
Add the corresponding pair of bits and the carry in
Produce a sum bit and a carry out
1 0 1 1
0 1 0 1
1 0 0 0 0
1
1 1
1
110. Subtracts LSD column in binary subtraction
HALF SUBTRACTOR
A
B
Di (difference)
B0 (borrow out)
Half
Subtractor
Input Output
Logic
Symbol:
Logic
Diagram:
111. Half Subtractor
C
A B D
0 0 0 1
0 0 0 0
0 1 1 1
1 0 1 0
1 1 0 0
A0
B0
D
0
C1
0
-1
1
2
1
112. Used for subtracting binary place
values other than the 1s place
FULL SUBTRACTOR
Logic
Symbol:
Logic
Diagram:
A
B
Di (difference)
B0 (borrow out)
Full
Subtractor
Input Output
Bin
A
B
Di
B0
H. S.
H. S.
Bin
113. Full Subtractor
0 0 0 0 0
0 0 1 1 1
0 1 0 1 0
0 1 1 0 0
1 0 0 1 1
1 0 1 0 1
1 1 0 0 0
1 1 1 1 1
Ci Ai Bi Di Ci+1
1 1
1 1
Ci
AiBi
00 01 11 10
0
1
Di
Di = Ci (Ai Bi)
Same as Si in full adder
114. Full Subtractor
0 0 0 0 0
0 0 1 1 1
0 1 0 1 0
0 1 1 0 0
1 0 0 1 1
1 0 1 0 1
1 1 0 0 0
1 1 1 1 1
Ci Ai Bi Di Ci+1 Ci
AiBi
00 01 11 10
0
1
1
1 1
1
Ci+1
Ci+1 = !Ai & Bi
# Ci & !Ai & !Bi
# Ci & Ai & Bi
115. Full Subtractor
Ci+1 = !Ai & Bi
# Ci & !Ai & !Bi
# Ci & Ai & Bi
Ci+1 = !Ai & Bi
# Ci & (!Ai & !Bi # Ai Bi)
Ci+1 = !Ai & Bi # Ci & !(Ai Bi)
Recall:
Di = Ci (Ai Bi)
Ci+1 = !Ai & Bi # Ci & !(Ai Bi)
119. BCD to Excess-3 Code
Conversion
Simplified functions
Z
Y
X
W
=
=
=
D'
CD +C'D'
B'C + B'D+BC'D'
= A+BC+BD
120. Decoder
An n-to-m decoder
n
a binary code of n bits = 2 distinct information
n input variables; up to 2 output lines
n
only one output can be active (high) at any time
122. Decoder with Enable
/Demultiplexer
Demultiplexers
a decoder with an enable input
receive information on a single line and transmits it on one of
n
2 possible output lines
0
Two-to-four-line decoder with enable input
124. 4x16 Decoder
Expansion
two 3-to-8 decoder: a 4-to-16 decoder
4 16 decoder
constructed with
3 8 decoders
two
125. Combinational Logic
Implementation
Each output = a minterm
Use a decoder and an external OR gate to implement any
Boolean function of
A full-adder
S(x,y,x)=(1,2,4,7)
C(x,y,z)= (3,5,6,7)
n input variables
126. Lan
Encoder
with three OR gates.
The encoder can be implemented
z D1 D3 D5 D7
y D2 D3 D6 D7
x D4 D5 D6 D7
128. Priority Encoder
Resolve the ambiguity of illegal inputs
Only one of the input is encoded
LSB MSB
D3 has the highest priority
the lowest priority
D0 has
X: don't-care conditions
V: valid output indicator
131. Multiplexer
Select binary information from one of many input
lines and direct it to a single output line
n
2 input lines, n selection lines and one output line
E.g.: 2-to-1-line multiplexer
Two-to-one-line multiplexer
133. Boolean Function
Implementation Using MUX
MUX: a decoder + an OR gate
2 -to-1 MUX can implement any Boolean function of n
input variable.
Procedure:
assign an ordering sequence of the input variable
the rightmost variable (D) will be used for the input lines
assign the remaining n-1 variables to the selection lines w.r.t.
their corresponding sequence
construct the truth table
n
consider a pair of consecutive
determine the input lines
minterms starting from m0
138. FLIP FLOPS
Characteristics
- 2 stable states
- Memory capability
- Operation is specified by a Characteristic Table
0-state 1-state
In order to be used in the computer circuits, state of the flip flop should
have input terminals and output terminals so that it can be set to a certain
state, and its state can be read externally.
R
S
Q
Q’
S R Q(t+1)
0 0 Q(t)
0 1 0
1 0 1
1 1 indeterminate
(forbidden)
Flip Flops
1 0 0 1
0 1 1 0
139. CLOCKED FLIP FLOPS
In a large digital system with many flip flops, operations of individual flip flops
are required to be synchronized to a clock pulse. Otherwise,
the operations of the system may be unpredictable.
R
S
Q
Q’
c
(clock)
Flip Flops
S Q
c
R Q’
S Q
c
R Q’
operates when operates when
clock is high clock is low
Clock pulse allows the flip flop to change state only
when there is a clock pulse appearing at the c terminal.
We call above flip flop a Clocked RS Latch, and symbolically as
140. D-LATCH
D-Latch
Forbidden input values are forced not to occur
by using an inverter between the inputs
Flip Flops
Q
Q’
D(data)
E
(enable)
D Q
E Q’
E Q’
D Q
D Q(t+1)
0 0
1 1
141. EDGE-TRIGGERED FLIP
FLOPS
Characteristics
- State transition occurs at the rising edge or
falling edge of the clock pulse
Latches
Edge-triggered Flip Flops (positive)
respond to the input only during these periods
respond to the input only at this time
Flip Flops
142. POSITIVE EDGE-TRIGGERED
T-Flip Flop: JK-Flip Flop whose J and K inputs are tied together to make
T input. Toggles whenever there is a pulse on T input.
Flip Flops
D-Flip Flop
JK-Flip Flop
S1 Q1
C1
R1 Q1'
S2 Q2
C2
R2 Q2'
D
C
Q
Q'
D
C
Q
Q'
SR1 SR2
SR1 active
SR2 active
D-FF
S1 Q1
C1
R1 Q1'
S2 Q2
C2
R2 Q2'
SR1 SR2
J
K
C
Q
Q'
J Q
C
K Q'
SR1 active
SR2 inactive SR2 inactive
SR1 inactive