Basics of Digital Systems. Boolean algebra Truth tables Karnaugh maps


 Merilyn Cooper
 2 years ago
 Views:
Transcription
1 Basics of Digital Systems Boolean algebra Truth tables Karnaugh maps
2 Boolean Algebra In digital systems we deal with the binary number system. This means that the value of an element can be either 0 or 1. The elements can also be referred to as false or true for 0 or 1 respectively. There are three operators in Boolean algebra: AND OR NOT
3 Boolean Algebra: AND The AND operator is by a dot between the two elements or nothing between the two elements at all. It is also know as the product of elements. x y or x y The AND operator requires that all elements have a value 1 (or true) for the result to be 1 (or true). Thus if we had two elements x, y, both x AND y need to be 1 for the result to be one. x y x y
4 Boolean Algebra: OR The OR operator is represented the same as addition in regular algebra, with a plus sign. It is also known as the sum of elements. x + y The OR operator requires that either of the two elements be 1 for the result to be 1. So if we had two element x, y, then x OR y has to have a value of 1 for the result to have a value of 1. x y x + y
5 Boolean Algebra: NOT The NOT operator is represented with an apostrophe after the element. It is also known as the complement of an element. x The NOT operator give the opposite of the element s value as the result. So for an element x with a value of 1, the NOT of x is equal to 0. x x
6 Boolean Algebra: Properties Boolean algebra is governed by several key properties. Commutative x + y = y + x x y = y x Associative x + (y + z) = (x + y) + z x (y z) = (x y) z Distributive x (y + z) = x y + x z x + y z = (x +y)(x + z) Absorption x + x y = x x (x + y) = x Involution (x ) = x DeMorgan (x + y) = x y (x y) = x + y
7 Boolean Algebra: Properties Using the previous properties and the knowledge of the different operators, we can find a few more important properties of Boolean algebra. x + 0 = x x 1 = x x + x = 1 x x = 0 x + x = x x x = x x + 1 = 1 x 0 = 0
8 Boolean Algebra: Simplifying Using the properties we can simplify Boolean expressions. For example lets simplify the following expression. F = x y z + x y z + x y First we can see the use of the distributive property. We can apply is to simplify the expression to: F = x z (y + y) + x y Next we can apply one of our derived properties. (x + x ) = 1. This gives us a final simplification of: F = x z + x y This simplification process can be much easier through the use of Truth Tables and Karnaugh Maps.
9 Minterms and Maxterms In addition to simplifying a boolean expression we can also expand them. There are two ways of expanding a boolean expression: a sum of minterms or a product of maxterms. The minterms are the products of each of the elements in the expression in every combination (the elemnts can be complmented or not. Ex. x y z ). Each minterm has a designation starting with m 0. The maxterms are the sums of each of the elements in the expression in every expression (the elements can be complements or not. Ex. x + y + z ). Each maxterm has a designation starting with M 0.
10 Minterms and Maxterms cont. The minterms and maxterms for an expression with 3 elements are as follows. Minterms Maxterms Term Designation Term Designation x y z m 0 x + y + z M 0 x y z m 1 x + y + z M 1 x y z m 2 x + y + z M 2 x y z m 3 x + y + z M 3 x y z m 4 x + y + z M 4 z y z m 5 x + y + z M 5 x y z m 6 x + y + z M 6 x y z m 7 x + y + z M 7
11 Minterms and Maxterms Example If we look at the simplified version of the previous example we can expand it to a sum of minterms or a product of maxterm. F = x z + x y To have a sum of minterms, we need to make each term have all three element (x, y and z). We know that, And we know that, (x z)(1) = x z (y + y ) = 1 So we can add the y element into that term. x z = x z(y + y )
12 Minterms and Maxterms: Example cont. We can multiply the term out to get: x z = x y z + x y z By using this same method we can expand the second term. x y = x y z + x y z So we find that, F = x z + x y = x y z + x y z + x y z + x y z We can express this as a sum of minterm designations. F = m 1 + m 3 + m 4 + m 5
13 Minterms and Maxterms: Example cont. Now we will expand our function to a product of maxterms. F = x z + x y This is easiest to do by taking the complement of the sum of minterms. F = (m 1 + m 3 + m 4 + m 5 ) F = m 1 m 3 m 4 m 5 F = (x y z) (x yz) (x y z) (x y z ) F = (x + y + z )(x + y + z )(x + y + z )(x + y + z) F = M 1 M 3 M 4 M 5 We now have our two expansions of our function. As a sum of minterms and as a product of maxterms.
14 Truth Tables A truth tables is a method of setting up and testing a boolean expression for every possible input. The truth table is set up with a column for each different element in the expression and an additional column for the fuction its self. For example: F = x + y z x y z F
15 Truth Tables cont. We then want to fill in the columns with our elements x, y and z to allow for every possible combination of inputs. As you may notice, each row corresponds to one minterm starting with m 0 and ending with m 7. x y z F
16 Truth Tables cont. Next we input each of these cases into the function and calculate what the output will be. The output goes under the function s column. x y z F
17 Karnaugh Map The truth table is great for visualizing for which inputs you get a 0 or 1, however it may not be easy to use it to simplify a boolean expression. For that we use the Karnaugh map. The Karnaugh map or Kmap is a table setup in such a way to make simplifying expressions really easy. It works with minterms like the truth tables, this makes going from one to the other fairly simple. We ll start my looking at a two variable map.
18 TwoVariable Kmap The first Kmap shows the layout for an expression with two elements. In the second Kmap we can see the relation between the minterms and the elements. m 0 m 1 y x x y x y m 2 m 3 1 x y x y Once we have the Kmap drawn, we can look at our truth table and in each box we place the output we found corresponding to each minterm.
19 TwoVariable Example Lets look at the the following function. F = x y + x y + x y We start by making a truth table. x y F Then we construct our Kmap using the truth table. y x
20 TwoVariable Example cont. In order to use this Kmap to reduce our expression we need to remember one of our properties of boolean algebra: (y + y ) = 1 If we use this with our minterms we see that, x y + x y = x (y + y) = x or m 0 + m 1 = x We can show this in our Kmap but highlighting those given boxes. y x
21 TwoVariable Example cont. We can do the same thing with m 0 and m 2. y x Using these highlightings we can reduce the expression to, F = x + y
22 ThreeVariable Kmap The three variable map can be extended from the two variable map. Instead of having only y along the top we put yz. m 0 m 1 m 3 m 2 y z x x y z x y z x y z x y z m 4 m 5 m 7 m 6 1 x y z x y z x y z x y z It is important to note that the minterms are NOT placed in ascending order. The third and fourth column are switched. If we look at the kmap with the elements, we can see that the columns are labeled like gray code. By having only one bit change between each row/column we are able to use our highlighting method.
23 ThreeVariable Example 1 Lets look are the following example, F = m 2 + m 3 + m 4 + m 5 y z x When highlighting with a three variable kmap we can highlight pairs of two, lines of four, boxes of four, or all eight. This is because we set up the kmap such that there is only one bit change from column to column or row to row (like gray code).
24 ThreeVariable Example 1 To figure out what the resulting simplification will be, look are each highlighted section and determine which bits don t change. In our example, for the yellow highlighting, we see that x stays at one and y stays at zero. Thus our simplification for that highlighting is, x y We do the same thing for the second highlighting and add them together to get, F = x y + x y
25 ThreeVariable Example 2 Now we will look at a more complicated example. F = m 0 + m 2 + m 4 + m 5 + m 6 y z x Before we highlight an important thing to note is that between m 0 and m 2, there is only a one bit change. This mean that when highlighting we are allowed to wrap around the edges of the table.
26 ThreeVariable Example 2 For our expression we can highlight a box around the edges as well as a pair on the bottom. y z x Again we look at the variables that don t change for each highlighting. For the box, x varies between 1 and 0, y varies between 1 and 0, and z stays at 0. So our expression for the red box is, z
27 ThreeVariable Example 2 Doing the same thing with the pair highlighting we get a final answer of, F = z + x y
28 FourVariable Kmap The four variable kmap functions the same as the three variable kmap, but we must make sure that there is only a one bit change between rows and columns. This means that in addition to the third and four column being switched, the third and fourth row are also switched. m 0 m 1 m 3 m 2 m 4 m 5 m 7 m 6 m 12 m 13 m 15 m 14 m 8 m 9 m 11 m 10 y z wx w x y z w x y z w x y z w x y z 01 w x y z w x y z w x y z w x y z 11 w x y z w x y z w x y z w x y z 10 w x y z w x y z w x y z w x y z
29 FourVariable Kmap Just like with the three variable map, we are allowed to wrap around the edges of the map, both on the sides and the top/bottom. For the four variable map there are many possible highlightings: a pair, a line of four, a box of four, a 2x4 box of eight (two full columns or rows), or all sixteen. As a special note, the for corners actually create a box of four because you can wrap both horiztontally and vertically.
30 FourVariable Example Lets use the follow function, F= m 0 + m 1 + m 2 + m 8 + m 9 + m 10 + m 15 y z wx For this example we have two boxes of four and one left over.
31 FourVariable Example We can highlight it as follows. y z wx By looking at which variables don t change, we can find the reduced expression for each highlingting.
32 FourVariable Example The red highlighting reduces to, x z The blue highlighting reduces to, x y The purple highlighting cannot be reduced, it stay as, w x y z Adding everything together we get, F = x z + x y + w x y z
33 Sample Problems Try making a truth table and kmap to reduce the following expressions. 1. F= m 3 + m 4 + m 6 + m 7 2. F= x z + x y + x y z + y z 3. F= Σ(0, 1, 2, 4, 5, 6, 8, 9, 12, 13, 14) 4. F= m 1 + m 2 + m 4 + m 7 + m 8 + m 10 + m 13 + m 15
34 Solutions 1. y z + x z 2. z + x y 3. y + w z + x z 4. w x z + x y z + w x z +x y z + w x y z + w x y z
35 Reference Digital Design, Fourth Edition M. Morris Mano, Michael D. Ciletti
Karnaugh Maps. Circuitwise, this leads to a minimal twolevel implementation
Karnaugh Maps Applications of Boolean logic to circuit design The basic Boolean operations are AND, OR and NOT These operations can be combined to form complex expressions, which can also be directly translated
More informationKarnaugh Map. Alternative way to Boolean Function Simplification. Karnaugh Map. Description of Kmap & Terminology
Alternative way to Boolean Function Simplification Karnaugh Map CIT 595 Spring 2010 Simplification of Boolean functions leads to simpler (and usually faster) digital circuits Simplifying Boolean functions
More informationSimplifying Logic Circuits with Karnaugh Maps
Simplifying Logic Circuits with Karnaugh Maps The circuit at the top right is the logic equivalent of the Boolean expression: f = abc + abc + abc Now, as we have seen, this expression can be simplified
More informationCSEE 3827: Fundamentals of Computer Systems. Standard Forms and Simplification with Karnaugh Maps
CSEE 3827: Fundamentals of Computer Systems Standard Forms and Simplification with Karnaugh Maps Agenda (M&K 2.32.5) Standard Forms ProductofSums (PoS) SumofProducts (SoP) converting between Minterms
More informationKarnaugh Maps & Combinational Logic Design. ECE 152A Winter 2012
Karnaugh Maps & Combinational Logic Design ECE 52A Winter 22 Reading Assignment Brown and Vranesic 4 Optimized Implementation of Logic Functions 4. Karnaugh Map 4.2 Strategy for Minimization 4.2. Terminology
More informationBasics of Digital Logic Design
CSE 675.2: Introduction to Computer Architecture Basics of Digital Logic Design Presentation D Study: B., B2, B.3 Slides by Gojko Babi From transistors to chips Chips from the bottom up: Basic building
More informationBOOLEAN ALGEBRA & LOGIC GATES
BOOLEAN ALGEBRA & LOGIC GATES Logic gates are electronic circuits that can be used to implement the most elementary logic expressions, also known as Boolean expressions. The logic gate is the most basic
More informationPoints Addressed in this Lecture. Standard form of Boolean Expressions. Lecture 5: Logic Simplication & Karnaugh Map
Points Addressed in this Lecture Lecture 5: Logic Simplication & Karnaugh Map Professor Peter Cheung Department of EEE, Imperial College London (Floyd 4.54.) (Tocci 4.4.5) Standard form of Boolean Expressions
More informationConversion between number systems:
Conversion between number systems: Radixr to decimal. Decimal to binary. Decimal to Radixr Binary to Octal Binary to Hex Binary arithmetic operations. Negative number representations. Switching Algebra
More information4 BOOLEAN ALGEBRA AND LOGIC SIMPLIFICATION
4 BOOLEAN ALGEBRA AND LOGIC SIMPLIFICATION BOOLEAN OPERATIONS AND EXPRESSIONS Variable, complement, and literal are terms used in Boolean algebra. A variable is a symbol used to represent a logical quantity.
More informationCSE140: Midterm 1 Solution and Rubric
CSE140: Midterm 1 Solution and Rubric April 23, 2014 1 Short Answers 1.1 True or (6pts) 1. A maxterm must include all input variables (1pt) True 2. A canonical product of sums is a product of minterms
More informationGateLevel Minimization
Chapter 3 GateLevel Minimization 3 Outline! Karnaugh Map Method! NAND and NOR Implementations! Other TwoLevel Implementations! ExclusiveOR Function! Hardware Description Language 32 Why Logic Minimization?!
More information2.0 Chapter Overview. 2.1 Boolean Algebra
Thi d t t d ith F M k 4 2 Boolean Algebra hapter Two Logic circuits are the basis for modern digital computer systems. To appreciate how computer systems operate you will need to understand digital logic
More informationCSE 220: Systems Fundamentals I Unit 7: Logic Gates; Digital Logic Design: Boolean Equations and Algebra
CSE 220: Systems Fundamentals I Unit 7: Logic Gates; Digital Logic Design: Boolean Equations and Algebra Logic Gates Logic gatesare simple digital circuits that take one or more binary inputs and produce
More informationBoolean Algebra Part 1
Boolean Algebra Part 1 Page 1 Boolean Algebra Objectives Understand Basic Boolean Algebra Relate Boolean Algebra to Logic Networks Prove Laws using Truth Tables Understand and Use First Basic Theorems
More informationThe equation for the 3input XOR gate is derived as follows
The equation for the 3input XOR gate is derived as follows The last four product terms in the above derivation are the four 1minterms in the 3input XOR truth table. For 3 or more inputs, the XOR gate
More information1. Digital Logic Circuits
1 Digital Logic ircuits 1. Digital Logic ircuits Many scientific, industrial and commercial advances have been made possible by the advent of computers. Digital Logic ircuits form the basis of any digital
More informationExclusive OR/Exclusive NOR (XOR/XNOR)
Exclusive OR/Exclusive NOR (XOR/XNOR) XOR and XNOR are useful logic functions. Both have two or more inputs. The truth table for two inputs is shown at right. a XOR b = 1 if and only if (iff) a b. a XNOR
More informationArithmeticlogic units
Arithmeticlogic units An arithmeticlogic unit, or ALU, performs many different arithmetic and logic operations. The ALU is the heart of a processor you could say that everything else in the CPU is there
More informationBasic Logic Gates Richard E. Haskell
BASIC LOGIC GATES 1 E Basic Logic Gates Richard E. Haskell All digital systems are made from a few basic digital circuits that we call logic gates. These circuits perform the basic logic functions that
More informationDESIGN AND ANALYSIS USING JK AND T FLIPFLOPS
Supplement to Logic and Computer Design Fundamentals 4th Edition DESIGN ND NLYSIS USING JK ND T FLIPFLOPS Selected topics not covered in the fourth edition of Logic and Computer Design Fundamentals are
More information4.203 Write the truth table for each of the following logic functions:
3e4.5 4.201 According to DeMorgan s theorem, the complement of X + Y Z is X Y +Z. Yet both functions are 1 for XYZ = 110. How can both a function and its complement be 1 for the same input combination?
More informationComputer Organization I. Lecture 8: Boolean Algebra and Circuit Optimization
Computer Organization I Lecture 8: Boolean Algebra and Circuit Optimization Overview The simplification from SOM to SOP and their circuit implementation Basics of Logic Circuit Optimization: Cost Criteria
More informationArithmetic Circuits Addition, Subtraction, & Multiplication
Arithmetic Circuits Addition, Subtraction, & Multiplication The adder is another classic design example which we are obliged look at. Simple decimal arithmetic is something which we rarely give a second
More informationAnalog & Digital Electronics Course No: PH218
Analog & Digital Electronics Course No: PH218 Lec29: Combinational Logic Modules Course Instructor: Dr. A. P. VAJPEYI Department of Physics, Indian Institute of Technology Guwahati, India 1 Combinational
More informationCombinational Logic Circuits
Chapter 2 Combinational Logic Circuits J.J. Shann Chapter Overview 21 Binary Logic and Gates 22 Boolean Algebra 23 Standard Forms 24 TwoLevel Circuit Optimization 25 Map Manipulation 補 充 資 料 :QuineMcCluskey
More informationA Little Perspective Combinational Logic Circuits
A Little Perspective Combinational Logic Circuits COMP 251 Computer Organization and Architecture Fall 2009 Motivating Example Recall our machine s architecture: A Simple ALU Consider an ALU that can perform
More informationl What have discussed up until now & why: l C Programming language l More lowlevel then Java. l Better idea about what s really going on.
CS211 Computer Architecture l Topics Digital Logic l Transistors (Design & Types) l Logic Gates l Combinational Circuits l KMaps Class Checkpoint l What have discussed up until now & why: l C Programming
More informationCSE140: Components and Design Techniques for Digital Systems
CSE4: Components and Design Techniques for Digital Systems Tajana Simunic Rosing What we covered thus far: Number representations Logic gates Boolean algebra Introduction to CMOS HW#2 due, HW#3 assigned
More informationTakeHome Exercise. z y x. Erik Jonsson School of Engineering and Computer Science. The University of Texas at Dallas
TakeHome Exercise Assume you want the counter below to count mod6 backward. That is, it would count 0543210, etc. Assume it is reset on startup, and design the wiring to make the counter count
More informationECE Digital Logic Design. Laboratory Manual
ECE 1315 Digital Logic Design Laboratory Manual Guide to Assembling your Circuits Dr. Fernando RíosGutiérrez Dr. Rocio AlbaFlores Dr. Chris Carroll Department of Electrical and Computer Engineering University
More informationCOMBINATIONAL LOGIC CIRCUITS
COMBINATIONAL LOGIC CIRCUITS 4.1 INTRODUCTION The digital system consists of two types of circuits, namely: (i) Combinational circuits and (ii) Sequential circuits A combinational circuit consists of logic
More informationBoolean Algebra. Boolean Algebra. Boolean Algebra. Boolean Algebra
2 Ver..4 George Boole was an English mathematician of XIX century can operate on logic (or Boolean) variables that can assume just 2 values: /, true/false, on/off, closed/open Usually value is associated
More informationCHAPTER 3 Boolean Algebra and Digital Logic
CHAPTER 3 Boolean Algebra and Digital Logic 3.1 Introduction 121 3.2 Boolean Algebra 122 3.2.1 Boolean Expressions 123 3.2.2 Boolean Identities 124 3.2.3 Simplification of Boolean Expressions 126 3.2.4
More informationCH3 Boolean Algebra (cont d)
CH3 Boolean Algebra (cont d) Lecturer: 吳 安 宇 Date:2005/10/7 ACCESS IC LAB v Today, you ll know: Introduction 1. Guidelines for multiplying out/factoring expressions 2. ExclusiveOR and Equivalence operations
More informationTutorial 5 Special Combinational Logic Circuit
Tutorial 5 Special Combinational Logic Circuit Question 1 a) What is the function of an adder circuit? b) A halfadder adds two binary bits, true or false? c) A halfadder has a sum output only, true or
More informationKarnaugh Maps. Example A B C X 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 1. each 1 here gives a minterm e.g.
Karnaugh Maps Yet another way of deriving the simplest Boolean expressions from behaviour. Easier than using algebra (which can be hard if you don't know where you're going). Example A B C X 0 0 0 0 0
More informationReading and construction of logic gates
Reading and construction of logic gates A Boolean function is an expression formed with binary variables, a binary variable can take a value of 1 or 0. Boolean function may be represented as an algebraic
More informationDigital Principles and System Design
PARTA Questions 1. Convert (10101100)2 into octal. 2. What is the important property of XS3 code? 3. What is the drawback of a serial adder compared to parallel adder? 4. Represent (10)10 in sign2 s
More informationCS61c: Representations of Combinational Logic Circuits
CS61c: Representations of Combinational Logic Circuits J. Wawrzynek October 12, 2007 1 Introduction In the previous lecture we looked at the internal details of registers. We found that every register,
More informationRita Lovassy. Digital Technics
Rita Lovassy Digital Technics Kandó Kálmán Faculty of Electrical Engineering Óbuda University Budapest, 2013 Preface Digital circuits address the growing need for computer networking communications in
More informationDigital Logic and Design (EEE241) Lecture
Digital Logic and Design (EEE241) Lecture Dr. M. G. Abbas Malik abbas.malik@ciitlahore.edu.pk edu Picture Source: http://www.vanoast.com/oldportfolio/digitaldesignlogoone%5ba%5d.jpg Previous lecture
More informationLogic Design 2013/9/5. Introduction. Logic circuits operate on digital signals
Introduction Logic Design Chapter 2: Introduction to Logic Circuits Logic circuits operate on digital signals Unlike continuous analog signals that have an infinite number of possible values, digital signals
More informationComp 150 Booleans and Digital Logic
Comp 150 Booleans and Digital Logic Recall the bool date type in Python has the two literals True and False and the three operations: not, and, or. The operations are defined by truth tables (see page
More informationWorking with combinational logic
Working with combinational logic Simplification twolevel simplification exploiting don t cares algorithm for simplification Logic realization twolevel logic and canonical forms realized with NNs and
More informationLecture 12: More on Registers, Multiplexers, Decoders, Comparators and Wot Nots
Lecture 12: More on Registers, Multiplexers, Decoders, Comparators and Wot Nots Registers As you probably know (if you don t then you should consider changing your course), data processing is usually
More informationDEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING
DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING Lab Manual Digital Electronics Laboratory (EC39) BACHELOR OF TECHNOLOGY Subject Code: EC 39 Subject Name: Digital Electronics Laboratory Teaching
More information3.Basic Gate Combinations
3.Basic Gate Combinations 3.1 TTL NAND Gate In logic circuits transistors play the role of switches. For those in the TTL gate the conducting state (on) occurs when the baseemmiter signal is high, and
More informationSum of Products (SOP) Expressions
Sum of Products (SOP) Expressions The Sum of Products (SOP) form of Boolean expressions and equations contains a list of terms (called minterms) in which all variables are ANDed (products). These minterms
More informationWorking with combinational logic. Design example: 2x2bit multiplier
Working with combinational logic Simplification twolevel simplification exploiting don t cares algorithm for simplification Logic realization twolevel logic and canonical forms realized with NNs and
More informationBasics of Digital Logic Design
Basics of Digital Logic Design Dr. Arjan Durresi Louisiana State University Baton Rouge, LA 70810 Durresi@Csc.LSU.Edu LSUEd These slides are available at: http://www.csc.lsu.edu/~durresi/csc3501_07/ Louisiana
More informationEE 110 Practice Problems for Exam 2: Solutions, Fall 2008
EE 110 Practice Problems for Exam 2: Solutions, Fall 2008 1. Circle T (true) or F (false) for each of these Boolean equations. (a). T FO An 8to1 multiplexer requires 2 select lines. (An 8to1 multiplexer
More informationDigital Design for Multiplication
Digital Design for Multiplication Norman Matloff October 15, 2003 c 2003, N.S. Matloff 1 Overview A cottage industry exists in developing fast digital logic to perform arithmetic computations. Fast addition,
More informationChapter 4 Boolean Algebra and Logic Simplification
ETEC 23 Programmable Logic Devices Chapter 4 Boolean Algebra and Logic Simplification Shawnee State University Department of Industrial and Engineering Technologies Copyright 27 by Janna B. Gallaher Boolean
More informationAlgebra Tiles Activity 1: Adding Integers
Algebra Tiles Activity 1: Adding Integers NY Standards: 7/8.PS.6,7; 7/8.CN.1; 7/8.R.1; 7.N.13 We are going to use positive (yellow) and negative (red) tiles to discover the rules for adding and subtracting
More information2.0 Chapter Overview. 2.1 Boolean Algebra
Thi d t t d ith F M k 4 0 2 Boolean Algebra Chapter Two Logic circuits are the basis for modern digital computer systems. To appreciate how computer systems operate you will need to understand digital
More information1. True or False? A voltage level in the range 0 to 2 volts is interpreted as a binary 1.
File: chap04, Chapter 04 1. True or False? A voltage level in the range 0 to 2 volts is interpreted as a binary 1. 2. True or False? A gate is a device that accepts a single input signal and produces one
More informationBinary Adders: Half Adders and Full Adders
Binary Adders: Half Adders and Full Adders In this set of slides, we present the two basic types of adders: 1. Half adders, and 2. Full adders. Each type of adder functions to add two binary bits. In order
More informationAljabar Boolean, Penyederhanaan Logika dan Peta Karnaugh
Aljabar Boolean, Penyederhanaan Logika dan Peta Karnaugh ENDY SA ENDY SA Program Studi Teknik Elektro Fakultas Teknik Universitas Muhammadiyah Prof. Dr. HAMKA Slide  6 1 Standard Forms of Boolean Expressions
More information1.10 (a) Effects of logic gates AND, OR, NOT on binary signals in a processor
Chapter 1.10 Logic Gates 1.10 (a) Effects of logic gates AND, OR, NOT on binary signals in a processor Microprocessors are the central hardware that runs computers. There are several components that make
More informationGrade 6 Math Circles. Multiplication
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Introduction Grade 6 Math Circles November 5/6, 2013 Multiplication At this point in your schooling you should all be very comfortable with multiplication.
More informationChapter 3: Combinational Logic Design
Chapter 3: Combinational Logic Design 1 Introduction We have learned all the prerequisite material: Truth tables and Boolean expressions describe functions Expressions can be converted into hardware circuits
More informationLAB 2: BOOLEAN THEOREMS
LAB 2: BOOLEAN THEOREMS OBJECTIVES 1. To implement DeMorgan's theorems in circuit simplification. 2. To design a combinational logic circuit with simplest logic gates representation using Karnaugh Mapping
More informationIf A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?
Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question
More informationOA310 Patterns in Addition Tables
OA310 Patterns in Addition Tables Pages 60 63 Standards: 3.OA.D.9 Goals: Students will identify and describe various patterns in addition tables. Prior Knowledge Required: Can add two numbers within 20
More informationENGR 1000, Introduction to Engineering Design. Counting in Binary
ENGR 1000, Introduction to Engineering Design Unit 1: Prerequisite Knowledge for Mechatronics Systems Lesson 1.1: Converting binary numbers to decimal numbers and back Objectives: Convert decimal numbers
More informationLogic in Computer Science: Logic Gates
Logic in Computer Science: Logic Gates Lila Kari The University of Western Ontario Logic in Computer Science: Logic Gates CS2209, Applied Logic for Computer Science 1 / 49 Logic and bit operations Computers
More informationIntroduction. Digital Logic Design 1. Simplifying Logic Circuits. SumofProducts Form. Algebraic Simplification
2007 Introduction BK TP.HCM Tran Ngoc Thinh HCMC University of Technology http://www.cse.hcmut.edu.vn/~tnthinh Basic logic gate functions will be combined in combinational logic circuits. Simplification
More informationEx. Convert the Boolean function F = x + y z into a sum of minterms by using a truth table.
Section 3.5  Minterms, Maxterms, Canonical Fm & Standard Fm Page 1 of 5 3.5 Canonical Fms In general, the unique algebraic expression f any Boolean function can be obtained from its truth table by using
More informationChapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 3 Additional Gates and Circuits Charles Kime & Thomas Kaminski 2008 Pearson Education, Inc. Overview Part 1 Gate Circuits
More informationBoolean Algebra (cont d) UNIT 3 BOOLEAN ALGEBRA (CONT D) Guidelines for Multiplying Out and Factoring. Objectives. Iris HuiRu Jiang Spring 2010
Boolean Algebra (cont d) 2 Contents Multiplying out and factoring expressions ExclusiveOR and ExclusiveNOR operations The consensus theorem Summary of algebraic simplification Proving validity of an
More informationL2: Combinational Logic Design (Construction and Boolean Algebra)
L2: Combinational Logic Design (Construction and oolean lgebra) cknowledgements: Materials in this lecture are courtesy of the following sources and are used with permission. Prof. Randy Katz (Unified
More informationCombinational logic lab
ECE2883 HP: Lab 3 Logic Experts (LEs) Combinational logic lab Implementing combinational logic with Quartus We should be starting to realize that you, the SMEs in this course, are just a specific type
More informationDigital Logic Circuits
Digital Logic Circuits Digital describes any system based on discontinuous data or events. Typically digital is computer data or electronic sampling of an analog signal. Computers are digital machines
More informationSolving a System of Equations
11 Solving a System of Equations 111 Introduction The previous chapter has shown how to solve an algebraic equation with one variable. However, sometimes there is more than one unknown that must be determined
More informationThe Storage or Data Register
The Storage or Data Register All sequential logic circuits in the computer CPU are based on the latch or flipflop. A significant part of the ALU is the register complement. In the MIPS R2000 computer
More informationChapter 4. Gates and Circuits. Chapter Goals. Chapter Goals. Computers and Electricity. Computers and Electricity. Gates
Chapter Goals Chapter 4 Gates and Circuits Identify the basic gates and describe the behavior of each Describe how gates are implemented using transistors Combine basic gates into circuits Describe the
More informationMATH 105: Finite Mathematics 26: The Inverse of a Matrix
MATH 05: Finite Mathematics 26: The Inverse of a Matrix Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006 Outline Solving a Matrix Equation 2 The Inverse of a Matrix 3 Solving Systems of
More informationReview of ECE 230 Material Prof. A. Mason, Michigan State University
Review of ECE 230 Material Prof. A. Mason, Michigan State University Preface This document was developed for students taking ECE 331 to review material covered in ECE 230. It will be assumed that ECE 331
More informationQuineMcClusky Minimization Procedure
QuineMcClusky Minimization Procedure This is basically a tabular method of minimization and as much it is suitable for computer applications. The procedure for optimization as follows: Step : Describe
More information(2 4 + 9)+( 7 4) + 4 + 2
5.2 Polynomial Operations At times we ll need to perform operations with polynomials. At this level we ll just be adding, subtracting, or multiplying polynomials. Dividing polynomials will happen in future
More informationMost decoders accept an input code and produce a HIGH ( or a LOW) at one and only one output line. In otherworlds, a decoder identifies, recognizes,
Encoders Encoder An encoder is a combinational logic circuit that essentially performs a reverse of decoder functions. An encoder accepts an active level on one of its inputs, representing digit, such
More informationIntroduction Number Systems and Conversion
UNIT 1 Introduction Number Systems and Conversion Objectives 1. Introduction The first part of this unit introduces the material to be studied later. In addition to getting an overview of the material
More informationDrawing a histogram using Excel
Drawing a histogram using Excel STEP 1: Examine the data to decide how many class intervals you need and what the class boundaries should be. (In an assignment you may be told what class boundaries to
More information3.2 Simplify the following Boolean functions, using threevariable maps: (ay F(x, y, z) = L(o, 1,5,7)
Answers to problems marked with ~,appear at the end of the book. 3.1'~ Simplify the following Boolean functions, using threevariable maps: (a) F(x, y, z) = L(o, 2,6,7) (b) F(x, y, z) = L(o, 1,2,3,7) 3.2
More informationTwolevel logic using NAND gates
CSE140: Components and Design Techniques for Digital Systems Two and Multilevel logic implementation Tajana Simunic Rosing 1 Twolevel logic using NND gates Replace minterm ND gates with NND gates Place
More information2 1 Implementation using NAND gates: We can write the XOR logical expression A B + A B using double negation as
Chapter 2 Digital Logic asics 2 Implementation using NND gates: We can write the XOR logical expression + using double negation as + = + = From this logical expression, we can derive the following NND
More informationIntro to Digital Logic Circuits
FORDHAM UNIVERSITY CISC 3593 Fordham College Lincoln Center Computer Organization Dept. of Computer and Info. Science Spring, 2011 Lab 1 Intro to Digital Logic Circuits 1 Introduction In this lab, the
More informationChapter 4 BOOLEAN ALGEBRA AND THEOREMS, MIN TERMS AND MAX TERMS
Chapter 4 BOOLEAN ALGEBRA AND THEOREMS, MIN TERMS AND MAX TERMS Lesson 5 BOOLEAN EXPRESSION, TRUTH TABLE and product of the sums (POSs) [MAXTERMS] 2 Outline POS two variables cases POS for three variable
More informationState Reduction and State Assignment Techniques. Derek Hildreth and Timothy Price Brigham Young University  Idaho
State Reduction and Assignment Running head: STATE REDUCTION AND ASSIGNMENT State Reduction and State Assignment Techniques Derek Hildreth and Timothy Price Brigham Young University  Idaho State Reduction
More informationDigital Design Using Digilent FPGA Boards VHDL / ActiveHDL Edition
Digital Design Using Digilent FPGA Boards VHDL / ActiveHDL Edition Table of Contents 1. Introduction 1 1.1 Background 1 1.2 Digital Logic 5 1.3 VHDL 8 2. Basic Logic Gates 9 2.1 Truth Tables and Logic
More informationPhiladelphia University Faculty of Information Technology Department of Computer Science  Semester, 2007/2008.
Philadelphia University Faculty of Information Technology Department of Computer Science  Semester, 2007/2008 Course Syllabus Course Title: Computer Logic Design Course Level: 1 Lecture Time: Course
More informationGates, Circuits, and Boolean Algebra
Gates, Circuits, and Boolean Algebra Computers and Electricity A gate is a device that performs a basic operation on electrical signals Gates are combined into circuits to perform more complicated tasks
More informationKarnaugh Maps (Kmap) Alternate representation of a truth table
Karnaugh Maps (Kmap) lternate representation of a truth table Red decimal = minterm value Note that is the MS for this minterm numbering djacent squares have distance = 1 Valuable tool for logic minimization
More informationChapter 5: Sequential Circuits (LATCHES)
Chapter 5: Sequential Circuits (LATCHES) Latches We focuses on sequential circuits, where we add memory to the hardware that we ve already seen Our schedule will be very similar to before: We first show
More informationChapter 1: Number Systems and Conversion
Chapter : Number Systems and Conversion.6 Subtract in binary. Place a over each column from which it was necessary to borrow. (a) (b) (c).7 dd the following numbers in binary using 2 s complement to represent
More informationGrade 9 Mathematics Unit #1 Number Sense SubUnit #1 Rational Numbers. with Integers Divide Integers
Page1 Grade 9 Mathematics Unit #1 Number Sense SubUnit #1 Rational Numbers Lesson Topic I Can 1 Ordering & Adding Create a number line to order integers Integers Identify integers Add integers 2 Subtracting
More informationSwitching Algebra and Logic Gates
Chapter 2 Switching Algebra and Logic Gates The word algebra in the title of this chapter should alert you that more mathematics is coming. No doubt, some of you are itching to get on with digital design
More informationCHAPTER TWO. 2.1 Unsigned Binary Counting. Numbering Systems
CHAPTER TWO Numbering Systems Chapter one discussed how computers remember numbers using transistors, tiny devices that act like switches with only two positions, on or off. A single transistor, therefore,
More informationSection P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities
Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.
More information