. T SHREE MAHAPRABHU PUBLIC SCHOOL & COLLEGE NOTES FOR BOARD EXAMINATION SUBJECT COMPUTER SCIENCE (Code: 083) Boolean Algebra
|
|
- Geraldine Ella Norris
- 5 years ago
- Views:
Transcription
1 . T SHREE MAHAPRABHU PUBLIC SCHOOL & COLLEGE NOTES FOR BOARD EXAMINATION SUBJECT COMPUTER SCIENCE (Code: 083) Boolean Algebra Introduction to Boolean Algebra Boolean algebra which deals with two-valued (true / false or 1 and 0) variables and functions find its use in modern digital computers since they too use two-level systems called binary systems. Let us examine the following statement:"i will buy a car If I get a salary increase or I win the lottery." This statement explains the fact that the proposition "buy a car" depends on two other propositions "get a salary increase" and "win the lottery". Any of these propositions can be either true or false hence the table of all possible situations: Salary Increase Win Lottery Buy a car = Salary Increase or Win Lottery False False False False True True True False True True True True The mathematician George Boole, hence the name Boolean algebra, used 1 for true, 0 for false and + for the or connective to write simpler tables. Let X = "get a salary increase", Y = "win the lottery" and F = "buy a car". The above table can be written in much simpler form as shown below and it defines the OR function. X Y F = X + Y
2 Let us now examine the following statement:"i will be able to read e-books online if I buy a computer and get an internet connection." The proposition "read e-books" depends on two other propositions "buy a computer" and "get an internet connection". Again using 1 for True, 0 for False, F = "read e- books", X = "buy a computer", Y = "get an internet connection" and use. for the connective and, we can write all possible situations using Boolean algebra as shown below. The above table can be written in much simpler form as shown below and it defines the AND function. X Y F = X. Y We have so far defined two operators: OR written as + and AND written.. The third operator in Boolean algebra is the NOT operator which inverts the input. Whose table is given below where NOT X is written as X'. X NOT X = X' The 3 operators are the basic operators used in Boolean
3 algebra and from which more complicated Boolean expressions may be written. Example: F = X. (Y + Z) Truth Tables Truth tables are a means of representing the results of a logic function using a table. They are constructed by defining all possible combinations of the inputs to a function, and then calculating the output for each combination in turn. For the three functions we have just defined, the truth tables are as follows. AND X Y F(X,Y) OR X Y F(X,Y) NOT X F(X) Truth tables may contain as many input variables as desired
4 F(X,Y,Z) = X.Y + Z X Y Z F(X,Y,Z) Different Properties or Laws of Boolean Algebra A "property" or a "law," describes how differing variables relate to each other in a system of numbers. Commutative Property It applies equally to addition and multiplication. In essence, the commutative property tells us we can reverse the order of variables that are either added together or multiplied together without changing the truth of the expression.
5 Associative Property This property tells us we can associate groups of added or multiplied variables together with parentheses without altering the truth of the equations.
6 Distributive Property Distributive Property, illustrating how to expand a Boolean expression formed by the product of a sum, and in reverse shows us how terms may be factored out of Boolean sums-ofproducts.
7 To summarize, here are the three basic properties: commutative, associative, and distributive. Identities In mathematics, an identity is a statement true for all possible values of its variable or variables. The algebraic identity of x + 0 = x tells us that anything (x) added to zero equals the original "anything," no matter what value that "anything" (x) may be. Boolean algebra has its own unique identities based on the bivalent states of Boolean variables. Inverse Another identity having to do with complementation is that of the double complement: a variable inverted twice. Complementing a variable twice (or any even number of times) results in the original Boolean value. This is analogous to negating (multiplying by -1) in real-number algebra: an even number of negations cancel to leave the original value.
8 Duality Principle In Boolean algebras the duality Principle can be is obtained by interchanging AND and OR operators and replacing 0's by 1's and 1's by 0's. Compare the identities on the left side with the identities on the right. Example X.Y+Z' = (X'+Y').Z Indempotent Law An input AND ed with itself or OR ed with itself is equal to that input. 1. A + A = A, A variable OR'ed with itself is always equal to the variable. 2. A. A = A, A variable AND'ed with itself is always equal to the variable. Involution Law: A =A When A=0, A =1, A =1 =0=A When A=1, A =0, A =0 =1=A Thus A =A Absorption Law: (i) A+AB=A LHS=A+AB=A.1+A.B=A(1+B)+A(B+1)=A.1=A=RHS (ii) A.(A+B)=A
9 LHS=A.)A+B)=A.A+A.B=A+A.B=A+A.B=A(1+B)=A.1=A=RHS Complementary Law A term ANDed with its complement equals 0, and a term ORed with its complement equals 1 AA' = 0 A+A' = 1 De Morgan s Theorem De Morgan was a great logician and Mathematician, as well as a friend of Charles Boole. The theorems given by De Morgan are associated with Boolean algebra. First Theorem: The complement of a sum equals to the product of the complements. (A+B) =A.B Proof: LHS= (A+B) = (0+0) = 0 =1 RHS=A.B =0.0 =1.1=1 Second Theorem: The complement of a product equals the sum of the complements. Proof: LHS = (A.B) = (0.0) = 0 = 1 RHS = A + B = =1 +1 =1 Logical Gates Logical Gates A logic gate is an elementary building block of a digital circuit. Most logic gates have two inputs and one output. At any given moment, every terminal is in one of the two binary conditions low (0) or high (1), represented by different voltage levels. The logic state of a terminal can, and generally does, change often, as the circuit processes data. In most logic gates, the low state is approximately zero volts (0 V), while the high state is approximately five volts positive (+5 V).
10 There are seven basic logic gates: AND, OR, XOR, NOT, NAND, NOR, and XNOR. The AND gate is so named because, if 0 is called "false" and 1 is called "true," the gate acts in the same way as the logical "and" operator. The following illustration and table show the circuit symbol and logic combinations for an AND gate. (In the symbol, the input terminals are at left and the output terminal is at right.) The output is "true" when both inputs are "true." Otherwise, the output is "false." AND gate Input 1 Input 2 Output The OR gate gets its name from the fact that it behaves after the fashion of the logical inclusive "or." The output is "true" if either or both of the inputs are "true." If both inputs are "false," then the output is "false." OR gate Input 1 Input 2 Output
11 1 1 The XOR ( exclusive-or ) gate acts in the same way as the logical "either/or." The output is "true" if either, but not both, of the inputs are "true." The output is "false" if both inputs are "false" or if both inputs are "true." Another way of looking at this circuit is to observe that the output is 1 if the inputs are different, but 0 if the inputs are the same. XOR gate Input 1 Input 2 Output A logical inverter, sometimes called a NOT gate to differentiate it from other types of electronic inverter devices, has only one input. It reverses the logic state. Inverter or NOT gate Input Output 1 The NAND gate operates as an AND gate followed by a NOT gate. It acts in the manner of the logical operation "and" followed by negation. The output is "false" if both inputs are "true." Otherwise, the output is "true." 1
12 NAND gate Input 1 Input 2 Output The NOR gate is a combination OR gate followed by an inverter. Its output is "true" if both inputs are "false." Otherwise, the output is "false." NOR gate Input 1 Input 2 Output The XNOR (exclusive-nor) gate is a combination XOR gate followed by an inverter. Its output is "true" if the inputs are the same, and"false" if the inputs are different. 1 XNOR gate
13 Input 1 1 Input 2 1 Output Using combinations of logic gates, complex operations can be performed. In theory, there is no limit to the number of gates that can be arrayed together in a single device. But in practice, there is a limit to the number of gates that can be packed into a given physical space. Arrays of logic gates are found in digital integrated circuits (ICs). As IC technology advances, the required physical volume for each individual logic gate decreases and digital devices of the same or smaller size become capable of performing ever-morecomplicated operations at ever-increasing speeds. Logic Circuits, Boolean Algebra, and Truth Tables Logic Representation There are three common ways in which to represent logic. 1. Truth Tables 2. Logic Circuit Diagram 3. Boolean Expression We will discuss each herein and demonstrate ways to convert between them. Truth Tables A truth table is a chart of 1s and 0s arranged to indicate the results (or outputs) of all possible inputs. The list of all possible inputs are arranged in columns on the left and the resulting outputs are listed in columns on the right. There are 2 to the power n possible states (or combination of inputs). For example with three inputs there are 2^3=8 possible combination of inputs.
14 Logic Diagram A logic diagram uses the pictoral description of logic gates in combination to represent a logic expression. An example below shows a logic diagram with three inputs (A, B, and C) and one output (Y). The interpretation of this will become clear in the following sections. Boolean Expression Boolean Algebra can be used to write a logic expression in equation form. There are a few symbols that you ll recognize but need to redefine. Note: Sometimes when the! is used to represent the NOT it is used before the letter and sometimes it is used after the letter. Care should be used so that you understand which method is being used! Below is an example boolean expression. In fact, it represents the same logic as the example logic circuit diagram above.
15 Converting from a Logic Circuit Diagram to a Truth Table This conversion is accomplished by selecting each state (or combination of inputs) one at a time, replacing the inputs with their respective values and figuring the value of each point through the circuit until the output is reached. The final output value for each state is then listed in the truth table next to the value of each input. Below is a logic circuit diagram with the input values. Study it carefully for an extended period of time, it is an animated image and the inputs and output will change every few seconds. Below are the results of the conversion in truth table form Converting Logic Circuit Diagrams to Boolean Expressions To convert from a logic circuit diagram to a boolean expression we start by listing our inputs at the correct place and process the inputs through the gates, one gate at a time, writing the result at each gate s output. The following is the resulting boolean expression of each of the gates.
16 And here is an example of the process being carried out. The fact that the result simplifies to the XOR is merely coincidental. Converting Truth Tables to Boolean Expressions There are two methods for converting truth tables to boolean expressions. The Sum of Products The Product of Sums
17 Converting Boolean Expressions to Logic Diagrams Converting boolean expressions to logic diagrams is the most challenging conversion on this page because it requires a very good understanding of order of operation. Below is the order of operations used in this conversion. Converting a Truth Table to a Logic Diagram The easiest way to accomplish this is to first convert the truth table to a boolean expression and then to a logic diagram. Application of Boolean Logic Applications of Boolean
18 Logic
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
Boolean algebra. Examples of these individual laws of Boolean, rules and theorems for Boolean algebra are given in the following table.
The Laws of Boolean Boolean algebra As well as the logic symbols 0 and 1 being used to represent a digital input or output, we can also use them as constants for a permanently Open or Closed circuit or
More informationDigital Logic (2) Boolean Algebra
Digital Logic (2) Boolean Algebra Boolean algebra is the mathematics of digital systems. It was developed in 1850 s by George Boole. We will use Boolean algebra to minimize logic expressions. Karnaugh
More information4 Switching Algebra 4.1 Axioms; Signals and Switching Algebra
4 Switching Algebra 4.1 Axioms; Signals and Switching Algebra To design a digital circuit that will perform a required function, it is necessary to manipulate and combine the various input signals in certain
More informationChapter 2: Switching Algebra and Logic Circuits
Chapter 2: Switching Algebra and Logic Circuits Formal Foundation of Digital Design In 1854 George Boole published An investigation into the Laws of Thoughts Algebraic system with two values 0 and 1 Used
More informationLogic Design. Chapter 2: Introduction to Logic Circuits
Logic Design Chapter 2: Introduction to Logic Circuits Introduction Logic circuits perform operation on digital signal Digital signal: signal values are restricted to a few discrete values Binary logic
More informationBinary Logic and Gates. Our objective is to learn how to design digital circuits.
Binary Logic and Gates Introduction Our objective is to learn how to design digital circuits. These circuits use binary systems. Signals in such binary systems may represent only one of 2 possible values
More informationEEE130 Digital Electronics I Lecture #4
EEE130 Digital Electronics I Lecture #4 - Boolean Algebra and Logic Simplification - By Dr. Shahrel A. Suandi Topics to be discussed 4-1 Boolean Operations and Expressions 4-2 Laws and Rules of Boolean
More informationBinary Logic and Gates
1 COE 202- Digital Logic Binary Logic and Gates Dr. Abdulaziz Y. Barnawi COE Department KFUPM 2 Outline Introduction Boolean Algebra Elements of Boolean Algebra (Binary Logic) Logic Operations & Logic
More informationCHAPTER1: Digital Logic Circuits Combination Circuits
CS224: Computer Organization S.KHABET CHAPTER1: Digital Logic Circuits Combination Circuits 1 PRIMITIVE LOGIC GATES Each of our basic operations can be implemented in hardware using a primitive logic gate.
More informationBOOLEAN LOGIC. By- Neha Tyagi PGT CS KV 5 Jaipur II Shift, Jaipur Region. Based on CBSE curriculum Class 11. Neha Tyagi, KV 5 Jaipur II Shift
BOOLEAN LOGIC Based on CBSE curriculum Class 11 By- Neha Tyagi PGT CS KV 5 Jaipur II Shift, Jaipur Region Neha Tyagi, KV 5 Jaipur II Shift Introduction Boolean Logic, also known as boolean algebra was
More informationChapter 2 (Lect 2) Canonical and Standard Forms. Standard Form. Other Logic Operators Logic Gates. Sum of Minterms Product of Maxterms
Chapter 2 (Lect 2) Canonical and Standard Forms Sum of Minterms Product of Maxterms Standard Form Sum of products Product of sums Other Logic Operators Logic Gates Basic and Multiple Inputs Positive and
More informationChapter 2. Boolean Algebra and Logic Gates
Chapter 2 Boolean Algebra and Logic Gates Basic Definitions A binary operator defined on a set S of elements is a rule that assigns, to each pair of elements from S, a unique element from S. The most common
More informationEECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits)
EECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits) September 5, 2002 John Wawrzynek Fall 2002 EECS150 Lec4-bool1 Page 1, 9/5 9am Outline Review of
More informationBOOLEAN ALGEBRA THEOREMS
OBJECTIVE Experiment 4 BOOLEAN ALGEBRA THEOREMS The student will be able to do the following: a. Identify the different Boolean Algebra Theorems and its properties. b. Plot circuits and prove De Morgan
More informationOutline. EECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits) Combinational Logic (CL) Defined
EECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits) January 30, 2003 John Wawrzynek Outline Review of three representations for combinational logic:
More informationDIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA
DIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA 1 Learning Objectives Understand the basic operations and laws of Boolean algebra. Relate these operations and laws to circuits composed of AND gates, OR gates, INVERTERS
More informationCombinational Logic Design Principles
Combinational Logic Design Principles Switching algebra Doru Todinca Department of Computers Politehnica University of Timisoara Outline Introduction Switching algebra Axioms of switching algebra Theorems
More informationE&CE 223 Digital Circuits & Systems. Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev
E&CE 223 Digital Circuits & Systems Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev 4 of 92 Section 2: Boolean Algebra & Logic Gates Major topics Boolean algebra NAND & NOR gates Boolean
More informationLogic Gate Level. Part 2
Logic Gate Level Part 2 Constructing Boolean expression from First method: write nonparenthesized OR of ANDs Each AND is a 1 in the result column of the truth table Works best for table with relatively
More informationCS 121 Digital Logic Design. Chapter 2. Teacher Assistant. Hanin Abdulrahman
CS 121 Digital Logic Design Chapter 2 Teacher Assistant Hanin Abdulrahman 1 2 Outline 2.2 Basic Definitions 2.3 Axiomatic Definition of Boolean Algebra. 2.4 Basic Theorems and Properties 2.5 Boolean Functions
More informationCombinational Logic. Review of Combinational Logic 1
Combinational Logic! Switches -> Boolean algebra! Representation of Boolean functions! Logic circuit elements - logic gates! Regular logic structures! Timing behavior of combinational logic! HDLs and combinational
More informationSwitches: basic element of physical implementations
Combinational logic Switches Basic logic and truth tables Logic functions Boolean algebra Proofs by re-writing and by perfect induction Winter 200 CSE370 - II - Boolean Algebra Switches: basic element
More informationLecture 3: Boolean Algebra
Lecture 3: Boolean Algebra Syed M. Mahmud, Ph.D ECE Department Wayne State University Original Source: Prof. Russell Tessier of University of Massachusetts Aby George of Wayne State University Overview
More informationChapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 1 Gate Circuits and Boolean Equations Chapter 2 - Part 1 2 Chapter 2 - Part 1 3 Chapter 2 - Part 1 4 Chapter 2 - Part
More informationGates and Flip-Flops
Gates and Flip-Flops Chris Kervick (11355511) With Evan Sheridan and Tom Power December 2012 On a scale of 1 to 10, how likely is it that this question is using binary?...4? What s a 4? Abstract The operation
More informationChapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 3 Additional Gates and Circuits Overview Part 1 Gate Circuits and Boolean Equations Binary Logic and Gates Boolean Algebra
More informationBoolean Algebra. The Building Blocks of Digital Logic Design. Section. Section Overview. Binary Operations and Their Representation.
Section 3 Boolean Algebra The Building Blocks of Digital Logic Design Section Overview Binary Operations (AND, OR, NOT), Basic laws, Proof by Perfect Induction, De Morgan s Theorem, Canonical and Standard
More informationUNIVERSITI TENAGA NASIONAL. College of Information Technology
UNIVERSITI TENAGA NASIONAL College of Information Technology BACHELOR OF COMPUTER SCIENCE (HONS.) FINAL EXAMINATION SEMESTER 2 2012/2013 DIGITAL SYSTEMS DESIGN (CSNB163) January 2013 Time allowed: 3 hours
More informationSchool of Computer Science and Electrical Engineering 28/05/01. Digital Circuits. Lecture 14. ENG1030 Electrical Physics and Electronics
Digital Circuits 1 Why are we studying digital So that one day you can design something which is better than the... circuits? 2 Why are we studying digital or something better than the... circuits? 3 Why
More informationE&CE 223 Digital Circuits & Systems. Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev. Section 2: Boolean Algebra & Logic Gates
Digital Circuits & Systems Lecture Transparencies (Boolean lgebra & Logic Gates) M. Sachdev 4 of 92 Section 2: Boolean lgebra & Logic Gates Major topics Boolean algebra NND & NOR gates Boolean algebra
More informationChapter 2: Princess Sumaya Univ. Computer Engineering Dept.
hapter 2: Princess Sumaya Univ. omputer Engineering Dept. Basic Definitions Binary Operators AND z = x y = x y z=1 if x=1 AND y=1 OR z = x + y z=1 if x=1 OR y=1 NOT z = x = x z=1 if x=0 Boolean Algebra
More informationXOR - XNOR Gates. The graphic symbol and truth table of XOR gate is shown in the figure.
XOR - XNOR Gates Lesson Objectives: In addition to AND, OR, NOT, NAND and NOR gates, exclusive-or (XOR) and exclusive-nor (XNOR) gates are also used in the design of digital circuits. These have special
More informationChapter-2 BOOLEAN ALGEBRA
Chapter-2 BOOLEAN ALGEBRA Introduction: An algebra that deals with binary number system is called Boolean Algebra. It is very power in designing logic circuits used by the processor of computer system.
More informationDigital Circuit And Logic Design I. Lecture 3
Digital Circuit And Logic Design I Lecture 3 Outline Combinational Logic Design Principles (). Introduction 2. Switching algebra 3. Combinational-circuit analysis 4. Combinational-circuit synthesis Panupong
More informationReview: Additional Boolean operations
Review: Additional Boolean operations Operation: NAND (NOT-AND) NOR (NOT-OR) XOR (exclusive OR) Expressions: (xy) = x + y (x + y) = x y x y = x y + xy Truth table: x y (xy) x y (x+y) x y x y 0 0 1 0 1
More informationUniversity of Technology
University of Technology Lecturer: Dr. Sinan Majid Course Title: microprocessors 4 th year معالجات دقيقة المرحلة الرابعة ھندسة الليزر والبصريات االلكترونية Lecture 3 & 4 Boolean Algebra and Logic Gates
More informationcontrol in out in out Figure 1. Binary switch: (a) opened or off; (b) closed or on.
Chapter 2 Digital Circuits Page 1 of 18 2. Digital Circuits Our world is an analog world. Measurements that we make of the physical objects around us are never in discrete units but rather in a continuous
More informationProf.Manoj Kavedia 2 Algebra
` Logic Gates and Boolean 2 Algebra Chapter-2 (Hours:06 Marks:14 )( 12064 Digital Techniques) Logic Gates And Boolean Algebra 2.1 Logical symbol, logical expression and truth table of AND, OR, NOT, NAND,
More informationLearning Objectives. Boolean Algebra. In this chapter you will learn about:
Ref. Page Slide /78 Learning Objectives In this chapter you will learn about: oolean algebra Fundamental concepts and basic laws of oolean algebra oolean function and minimization Logic gates Logic circuits
More informationEvery time has a value associated with it, not just some times. A variable can take on any value within a range
Digital Logic Circuits Binary Logic and Gates Logic Simulation Boolean Algebra NAND/NOR and XOR gates Decoder fundamentals Half Adder, Full Adder, Ripple Carry Adder Analog vs Digital Analog Continuous»
More informationImplementation of Boolean Logic by Digital Circuits
Implementation of Boolean Logic by Digital Circuits We now consider the use of electronic circuits to implement Boolean functions and arithmetic functions that can be derived from these Boolean functions.
More informationWeek-I. Combinational Logic & Circuits
Week-I Combinational Logic & Circuits Overview Binary logic operations and gates Switching algebra Algebraic Minimization Standard forms Karnaugh Map Minimization Other logic operators IC families and
More informationChapter 2 Boolean Algebra and Logic Gates
Chapter 2 Boolean Algebra and Logic Gates Huntington Postulates 1. (a) Closure w.r.t. +. (b) Closure w.r.t.. 2. (a) Identity element 0 w.r.t. +. x + 0 = 0 + x = x. (b) Identity element 1 w.r.t.. x 1 =
More informationIn Module 3, we have learned about Exclusive OR (XOR) gate. Boolean Expression AB + A B = Y also A B = Y. Logic Gate. Truth table
Module 8 In Module 3, we have learned about Exclusive OR (XOR) gate. Boolean Expression AB + A B = Y also A B = Y Logic Gate Truth table A B Y 0 0 0 0 1 1 1 0 1 1 1 0 In Module 3, we have learned about
More informationChapter 2: Boolean Algebra and Logic Gates
Chapter 2: Boolean Algebra and Logic Gates Mathematical methods that simplify binary logics or circuits rely primarily on Boolean algebra. Boolean algebra: a set of elements, a set of operators, and a
More informationDIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA
DIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA 1 Learning Objectives Understand the basic operations and laws of Boolean algebra. Relate these operations and laws to circuits composed of AND gates, OR gates, INVERTERS
More informationMC9211 Computer Organization
MC92 Computer Organization Unit : Digital Fundamentals Lesson2 : Boolean Algebra and Simplification (KSB) (MCA) (29-2/ODD) (29 - / A&B) Coverage Lesson2 Introduces the basic postulates of Boolean Algebra
More informationContents. Chapter 2 Digital Circuits Page 1 of 30
Chapter 2 Digital Circuits Page 1 of 30 Contents Contents... 1 2 Digital Circuits... 2 2.1 Binary Numbers... 2 2.2 Binary Switch... 4 2.3 Basic Logic Operators and Logic Expressions... 5 2.4 Truth Tables...
More informationBOOLEAN ALGEBRA INTRODUCTION SUBSETS
BOOLEAN ALGEBRA M. Ragheb 1/294/2018 INTRODUCTION Modern algebra is centered around the concept of an algebraic system: A, consisting of a set of elements: ai, i=1, 2,, which are combined by a set of operations
More informationXI STANDARD [ COMPUTER SCIENCE ] 5 MARKS STUDY MATERIAL.
2017-18 XI STANDARD [ COMPUTER SCIENCE ] 5 MARKS STUDY MATERIAL HALF ADDER 1. The circuit that performs addition within the Arithmetic and Logic Unit of the CPU are called adders. 2. A unit that adds two
More informationCSE20: Discrete Mathematics for Computer Science. Lecture Unit 2: Boolan Functions, Logic Circuits, and Implication
CSE20: Discrete Mathematics for Computer Science Lecture Unit 2: Boolan Functions, Logic Circuits, and Implication Disjunctive normal form Example: Let f (x, y, z) =xy z. Write this function in DNF. Minterm
More informationWEEK 2.1 BOOLEAN ALGEBRA
WEEK 2.1 BOOLEAN ALGEBRA 1 Boolean Algebra Boolean algebra was introduced in 1854 by George Boole and in 1938 was shown by C. E. Shannon to be useful for manipulating Boolean logic functions. The postulates
More informationUnit 4: Computer as a logic machine
Unit 4: Computer as a logic machine Propositional logic Boolean algebra Logic gates Computer as a logic machine: symbol processor Development of computer The credo of early AI Reference copyright c 2013
More informationBoolean Algebra CHAPTER 15
CHAPTER 15 Boolean Algebra 15.1 INTRODUCTION Both sets and propositions satisfy similar laws, which are listed in Tables 1-1 and 4-1 (in Chapters 1 and 4, respectively). These laws are used to define an
More informationCOS 140: Foundations of Computer Science
COS 140: Foundations of Computer Science Boolean Algebra Fall 2018 Introduction 3 Problem................................................................. 3 Boolean algebra...........................................................
More informationCS 226: Digital Logic Design
CS 226: Digital Logic Design 0 1 1 I S 0 1 0 S Department of Computer Science and Engineering, Indian Institute of Technology Bombay. 1 of 29 Objectives In this lecture we will introduce: 1. Logic functions
More informationBoolean Algebra and Logic Gates
Boolean Algebra and Logic Gates ( 范倫達 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C. Fall, 2017 ldvan@cs.nctu.edu.tw http://www.cs.nctu.edu.tw/~ldvan/ Outlines Basic
More informationLecture 2. Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits. Reading (Epp s textbook)
Lecture 2 Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits Reading (Epp s textbook) 2.1-2.4 1 Logic Logic is a system based on statements. A statement (or
More informationUnit 2 Boolean Algebra
Unit 2 Boolean Algebra 1. Developed by George Boole in 1847 2. Applied to the Design of Switching Circuit by Claude Shannon in 1939 Department of Communication Engineering, NCTU 1 2.1 Basic Operations
More informationDigital Logic. Lecture 5 - Chapter 2. Outline. Other Logic Gates and their uses. Other Logic Operations. CS 2420 Husain Gholoom - lecturer Page 1
Lecture 5 - Chapter 2 Outline Other Logic Gates and their uses Other Logic Operations CS 2420 Husain Gholoom - lecturer Page 1 Digital logic gates CS 2420 Husain Gholoom - lecturer Page 2 Buffer A buffer
More informationFunctions. Computers take inputs and produce outputs, just like functions in math! Mathematical functions can be expressed in two ways:
Boolean Algebra (1) Functions Computers take inputs and produce outputs, just like functions in math! Mathematical functions can be expressed in two ways: An expression is finite but not unique f(x,y)
More informationBoolean Algebra & Logic Gates. By : Ali Mustafa
Boolean Algebra & Logic Gates By : Ali Mustafa Digital Logic Gates There are three fundamental logical operations, from which all other functions, no matter how complex, can be derived. These Basic functions
More informationBoolean Algebra and Digital Logic
All modern digital computers are dependent on circuits that implement Boolean functions. We shall discuss two classes of such circuits: Combinational and Sequential. The difference between the two types
More informationUnit 2 Boolean Algebra
Unit 2 Boolean Algebra 2.1 Introduction We will use variables like x or y to represent inputs and outputs (I/O) of a switching circuit. Since most switching circuits are 2 state devices (having only 2
More informationNumber System. Decimal to binary Binary to Decimal Binary to octal Binary to hexadecimal Hexadecimal to binary Octal to binary
Number System Decimal to binary Binary to Decimal Binary to octal Binary to hexadecimal Hexadecimal to binary Octal to binary BOOLEAN ALGEBRA BOOLEAN LOGIC OPERATIONS Logical AND Logical OR Logical COMPLEMENTATION
More informationComputer organization
Computer organization Levels of abstraction Assembler Simulator Applications C C++ Java High-level language SOFTWARE add lw ori Assembly language Goal 0000 0001 0000 1001 0101 Machine instructions/data
More informationEx: Boolean expression for majority function F = A'BC + AB'C + ABC ' + ABC.
Boolean Expression Forms: Sum-of-products (SOP) Write an AND term for each input combination that produces a 1 output. Write the input variable if its value is 1; write its complement otherwise. OR the
More informationEEA051 - Digital Logic 數位邏輯 吳俊興高雄大學資訊工程學系. September 2004
EEA051 - Digital Logic 數位邏輯 吳俊興高雄大學資訊工程學系 September 2004 Boolean Algebra (formulated by E.V. Huntington, 1904) A set of elements B={0,1} and two binary operators + and Huntington postulates 1. Closure
More information1. Name the person who developed Boolean algebra
MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 67 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR TH 2TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA
More informationComputer Organization: Boolean Logic
Computer Organization: Boolean Logic Representing and Manipulating Data Last Unit How to represent data as a sequence of bits How to interpret bit representations Use of levels of abstraction in representing
More informationNumber System conversions
Number System conversions Number Systems The system used to count discrete units is called number system. There are four systems of arithmetic which are often used in digital electronics. Decimal Number
More informationBoolean Algebra and Logic Gates Chapter 2. Topics. Boolean Algebra 9/21/10. EECE 256 Dr. Sidney Fels Steven Oldridge
Boolean Algebra and Logic Gates Chapter 2 EECE 256 Dr. Sidney Fels Steven Oldridge Topics DefiniGons of Boolean Algebra Axioms and Theorems of Boolean Algebra two valued Boolean Algebra Boolean FuncGons
More informationChapter 2 Boolean Algebra and Logic Gates
CSA051 - Digital Systems 數位系統導論 Chapter 2 Boolean Algebra and Logic Gates 吳俊興國立高雄大學資訊工程學系 Chapter 2. Boolean Algebra and Logic Gates 2-1 Basic Definitions 2-2 Axiomatic Definition of Boolean Algebra 2-3
More informationLogic Gates and Boolean Algebra
Logic Gates and oolean lgebra The ridge etween Symbolic Logic nd Electronic Digital Computing Compiled y: Muzammil hmad Khan mukhan@ssuet.edu.pk asic Logic Functions and or nand nor xor xnor not 2 Logic
More informationTheorem/Law/Axioms Over (.) Over (+)
material prepared by: MUKESH OHR Follow me on F : http://www.facebook.com/mukesh.sirji4u OOLEN LGER oolean lgebra is a set of rules, laws and theorems by which logical operations can be mathematically
More informationAdditional Gates COE 202. Digital Logic Design. Dr. Muhamed Mudawar King Fahd University of Petroleum and Minerals
Additional Gates COE 202 Digital Logic Design Dr. Muhamed Mudawar King Fahd University of Petroleum and Minerals Presentation Outline Additional Gates and Symbols Universality of NAND and NOR gates NAND-NAND
More informationof Digital Electronics
26 Digital Electronics 729 Digital Electronics 26.1 Analog and Digital Signals 26.3 Binary Number System 26.5 Decimal to Binary Conversion 26.7 Octal Number System 26.9 Binary-Coded Decimal Code (BCD Code)
More informationCHAPTER 3 LOGIC GATES & BOOLEAN ALGEBRA
CHPTER 3 LOGIC GTES & OOLEN LGER C H P T E R O U T C O M E S Upon completion of this chapter, student should be able to: 1. Describe the basic logic gates operation 2. Construct the truth table for basic
More informationBoolean Algebra. Boolean Variables, Functions. NOT operation. AND operation. AND operation (cont). OR operation
oolean lgebra asic mathematics for the study of logic design is oolean lgebra asic laws of oolean lgebra will be implemented as switching devices called logic gates. Networks of Logic gates allow us to
More informationCHAPTER 2 BOOLEAN ALGEBRA
CHAPTER 2 BOOLEAN ALGEBRA This chapter in the book includes: Objectives Study Guide 2.1 Introduction 2.2 Basic Operations 2.3 Boolean Expressions and Truth Tables 2.4 Basic Theorems 2.5 Commutative, Associative,
More informationELCT201: DIGITAL LOGIC DESIGN
ELCT2: DIGITAL LOGIC DESIGN Dr. Eng. Haitham Omran, haitham.omran@guc.edu.eg Dr. Eng. Wassim Alexan, wassim.joseph@guc.edu.eg Lecture 2 Following the slides of Dr. Ahmed H. Madian ذو الحجة 438 ه Winter
More informationDiscrete Mathematics. CS204: Spring, Jong C. Park Computer Science Department KAIST
Discrete Mathematics CS204: Spring, 2008 Jong C. Park Computer Science Department KAIST Today s Topics Combinatorial Circuits Properties of Combinatorial Circuits Boolean Algebras Boolean Functions and
More information10/14/2009. Reading: Hambley Chapters
EE40 Lec 14 Digital Signal and Boolean Algebra Prof. Nathan Cheung 10/14/2009 Reading: Hambley Chapters 7.1-7.4 7.4 Slide 1 Analog Signals Analog: signal amplitude is continuous with time. Amplitude Modulated
More informationDigital electronics form a class of circuitry where the ability of the electronics to process data is the primary focus.
Chapter 2 Digital Electronics Objectives 1. Understand the operation of basic digital electronic devices. 2. Understand how to describe circuits which can process digital data. 3. Understand how to design
More informationDigital Techniques. Figure 1: Block diagram of digital computer. Processor or Arithmetic logic unit ALU. Control Unit. Storage or memory unit
Digital Techniques 1. Binary System The digital computer is the best example of a digital system. A main characteristic of digital system is its ability to manipulate discrete elements of information.
More informationBoolean Algebra, Gates and Circuits
Boolean Algebra, Gates and Circuits Kasper Brink November 21, 2017 (Images taken from Tanenbaum, Structured Computer Organization, Fifth Edition, (c) 2006 Pearson Education, Inc.) Outline Last week: Von
More informationBoolean Algebra & Digital Logic
Boolean Algebra & Digital Logic Boolean algebra was developed by the Englishman George Boole, who published the basic principles in the 1854 treatise An Investigation of the Laws of Thought on Which to
More informationDigital Logic Design ABC. Representing Logic Operations. Dr. Kenneth Wong. Determining output level from a diagram. Laws of Boolean Algebra
Digital Logic Design ENGG1015 1 st Semester, 2011 Representing Logic Operations Each function can be represented equivalently in 3 ways: Truth table Boolean logic expression Schematics Truth Table Dr.
More informationLogic and Boolean algebra
Computer Mathematics Week 7 Logic and Boolean algebra College of Information Science and Engineering Ritsumeikan University last week coding theory channel coding information theory concept Hamming distance
More informationBOOLEAN ALGEBRA TRUTH TABLE
BOOLEAN ALGEBRA TRUTH TABLE Truth table is a table which represents all the possible values of logical variables / statements along with all the possible results of the given combinations of values. Eg:
More informationLOGIC GATES. Basic Experiment and Design of Electronics. Ho Kyung Kim, Ph.D.
Basic Eperiment and Design of Electronics LOGIC GATES Ho Kyung Kim, Ph.D. hokyung@pusan.ac.kr School of Mechanical Engineering Pusan National University Outline Boolean algebra Logic gates Karnaugh maps
More informationBinary addition example worked out
Binary addition example worked out Some terms are given here Exercise: what are these numbers equivalent to in decimal? The initial carry in is implicitly 0 1 1 1 0 (Carries) 1 0 1 1 (Augend) + 1 1 1 0
More informationSet Theory Basics of Set Theory. mjarrar Watch this lecture and download the slides
9/6/17 Birzeit University Palestine 2015 6.1. Basics of 6.2 Properties of Sets and Element Argument 6.3 Algebraic Proofs mjarrar 2015 1 Watch this lecture and download the slides Course Page: http://www.jarrar.info/courses/dmath/
More informationII. COMBINATIONAL LOGIC DESIGN. - algebra defined on a set of 2 elements, {0, 1}, with binary operators multiply (AND), add (OR), and invert (NOT):
ENGI 386 Digital Logic II. COMBINATIONAL LOGIC DESIGN Combinational Logic output of digital system is only dependent on current inputs (i.e., no memory) (a) Boolean Algebra - developed by George Boole
More informationChapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 1 Gate Circuits and Boolean Equations Charles Kime & Thomas Kaminski 2008 Pearson Education, Inc. (Hyperlinks are active
More informationCombinational Logic Fundamentals
Topic 3: Combinational Logic Fundamentals In this note we will study combinational logic, which is the part of digital logic that uses Boolean algebra. All the concepts presented in combinational logic
More informationChapter 3 Combinational Logic Design
Logic and Computer Design Fundamentals Chapter 3 Combinational Logic Design Part 1- Implementation Technology and Logic Design Overview Part 1-Implementation Technology and Logic Design Design Concepts
More informationEECS150 - Digital Design Lecture 19 - Combinational Logic Circuits : A Deep Dive
EECS150 - Digital Design Lecture 19 - Combinational Logic Circuits : A Deep Dive March 30, 2010 John Wawrzynek Spring 2010 EECS150 - Lec19-cl1 Page 1 Boolean Algebra I (Representations of Combinational
More informationCHAPTER III BOOLEAN ALGEBRA
CHAPTER III- CHAPTER III CHAPTER III R.M. Dansereau; v.. CHAPTER III-2 BOOLEAN VALUES INTRODUCTION BOOLEAN VALUES Boolean algebra is a form of algebra that deals with single digit binary values and variables.
More informationChapter 2. Digital Logic Basics
Chapter 2 Digital Logic Basics 1 2 Chapter 2 2 1 Implementation using NND gates: We can write the XOR logical expression B + B using double negation as B+ B = B+B = B B From this logical expression, we
More information