Boolean Algebra (cont d) UNIT 3 BOOLEAN ALGEBRA (CONT D) Guidelines for Multiplying Out and Factoring. Objectives. Iris HuiRu Jiang Spring 2010


 Roxanne Sparks
 6 years ago
 Views:
Transcription
1 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 equation Reading Unit 3 UNIT 3 BOOLEAN ALGEBRA (CONT D) Iris HuiRu Jiang Spring 2010 Objectives Guidelines for Multiplying Out and Factoring 3 4 In this unit, you will Continue to learn Boolean algebra Simplify, complement, multiply out and factor an expression Prove any theorem in an algebraic way or using a truth table Learn XOR Use 1. X(Y + Z) = XY + XZ 2. (X + Y)(X + Z) = X + YZ 3. (X + Y)(X' + Z) = XZ + X'Y For multiplying out, apply 2. and 3. before 1. to avoid unnecessary terms For factoring, apply 1., 2., 3. from right terms to left terms
2 Proof? Multiplying Out 1. X(Y + Z) = XY + XZ 2. (X + Y)(X + Z) = X + YZ 3. (X + Y)(X' + Z) = XZ + X'Y 5 1. (X + Y)(X + Z) = X + YZ (X + Y)(X + Z) = XX + XZ + YX + YZ = X + XZ + XY + YZ = X + XY + YZ = X + YZ Or, X = 0, (0 + Y)(0 + Z) = YZ 0 + YZ = YZ X = 1, (1 + Y)(1 + Z) = 11 = YZ = 1 2. (X + Y)(X' + Z) = XZ + X'Y (X + Y)(X' + Z) = XX' + XZ + YX' + YZ = 0 + XZ + X'Y + YZ =XZ+X'Y+(X' + X)YZ =XZ+X'Y+ X'YZ + XYZ expand & simplify =XZ(1 + Y) + X'Y(1 + Z) = XZ + X'Y 6 Multiplying out: POS SOP Example: (A + B + C')(A + B + D)(A + B + E)(A + D' + E)(A' + C) = (A + B + C'D)(A + B + E)[AC + A'(D' + E)] 2. = (A + B + C'DE)(AC + A'D' + A'E) = AC + ABC + A'BD' + A'BE + A'C'DE = AC + A'BD' + A'BE + A'C'DE By brute force 162 terms! Factoring 1. X(Y + Z) = XY + XZ 2. (X + Y)(X + Z) = X + YZ 3. (X + Y)(X' + Z) = XZ + X'Y 7 Factoring: SOP POS Example: AC + A'BD' + A'BE + A'C'DE = A C + A'(BD' + BE + C'DE) 3. = X Z + X' Y = (X + Y)(X' + Z) = (A + BD' + BE + C'DE)(A' + C) = [A + C'DE + B(D' + E)](A' + C) 2. = [ X + YZ ](A' + C) = (X + Y)(X + Z)(A' + C) = (A + C'DE + B)(A + C'DE + D' + E)(A' + C) 2. = (A + B + C')(A + B + D)(A + B + E)(A + D' + E)(A' + C) 8 XOR and XNOR Difference and equivalence
3 Operation ExclusiveOR ExclusiveOR Operations 9 ExclusiveOR (XOR) ( ) 0 0 = 0, 0 1 = 1, 1 0 = 1, 1 1 = 0 Symbol X Y X Y Boolean expression: X Y = X'Y + XY' Truth table X Y X Y check difference of inputs show the condition to make output ==1 10 X Y = X'Y + XY' Useful theorems: X 0 = X X Y = Y X (commutative) X 1 = X' (X Y) Z = X (Y Z) =X Y Z(associative) X X = 0 X(Y Z) =XY XZ (distributive) X X' = 1 (X Y)' = X Y' = X' Y = X'Y' + XY Prove distributive law? XY XZ = XY(XZ)' + (XY)'XZ (by definition) = XY(X' + Z') + (X' + Y')XZ (DeMorgan) = XYZ' + XY'Z = X(YZ' + Y'Z) = X(Y Z) (difference)' == equivalence Operation ExclusiveNOR Simplification of XOR and XNOR 11 Boolean expression: X Y = X'Y' + XY ExclusiveNOR (XNOR) () check 0 0 = 1, 0 1 = 0, 1 0 = 0, 1 1 = 1 equivalence Symbol Truth table of inputs X X Y X Y (XY) X Y Y X (X Y)' = (X Y) (difference)' == Y equivalence show the condition to make output ==1 12 X Y = X'Y + XY' X Y = X'Y' + XY (X'Y + XY')' = X'Y' + XY F = (A'B C) + (B AC') = [A'BC + (A'B)'C'] +[B'AC' + B(AC')'] = A'BC + (A + B')C' + B'AC' + B(A' + C) = B(A'C + A' + C) + C'(A + B' + AB') = B(A' + C) + C'(A + B') (can be further simplified!) F = A' B C = (A'B' + AB) C =(A'B' + AB)C' +(A'B' + AB)'C = A'B'C' + ABC' + A'BC + AB'C
4 The Consensus Theorem 13 The Consensus Theorem Redundancy removal 14 XY + X'Z + YZ = XY + X'Z (YZ is redundant) Proof: XY + X'Z + YZ = XY + X'Z + (X + X')YZ = (XY + XYZ) + (X'Z + X'YZ) = XY(1 + Z) + X'Z(1 + Y) = XY + X'Z How to find consensus terms? 1. Find a pair of terms, one of which contains a variable and the other its complement A'C'D + A'BD + BCD + ABC + ACD' (A A') 2. Ignore the variable and its complement; the left variables composite the consensus term (A'BD ) + (ABC) BDBC = BCD (redundant term) a'b'+ ac + bc' + b'c + ab = a'b'+ ac + bc' Ordering Does Matter! Dual Form of the Consensus Theorem A'C'D + A'BD + BCD + ABC + ACD' A'C'D + A'BD + BCD + ABC + ACD' = A'C'D + A'BD + ABC + ACD' 4 terms A'C'D + A'BD + BCD + ABC + ACD' XY + X'Z + YZ = XY + X'Z YZ is redundant dual (X + Y)(X' + Z)(Y + Z) = (X + Y)(X' + Z) (Y + Z) is redundant (a + b + c')(a + b + d')(b + c + d') = (a + b + c')(b + c + d') = A'C'D + BCD + ACD' Only 3 terms! (a + b + c') + (b + c + d') a + b + b + d' = a + b + d'
5 Redundancy Injection 17 ABCD + B'CDE + A'B' + BCE' Find consensus terms ABCD + B'CDE ACDE ABCD + A'B' BCD(B') + (ACD)A' B'CDE + BCE' CDE(CE') + B'CD(BC) A'B' + BCE' A'CE' Add consensus term ACDE 18 Summary of Algebraic Simplification Simplification can reduce cost ABCD + B'CDE + A'B' + BCE' + ACDE = A'B' + BCE' + ACDE Rule A  Combining Terms Rule B  Eliminating Terms XY + XY' = X(Y + Y') = X abc'd' + abcd' = abd' (X = abd', Y = c) ab'c + abc + a'bc = ab'c + abc + abc + a'bc (repeat term) = ac + bc (a + bc)(d + e') + a'(b' + c')(d + e') = d + e' (why?? DeMorgan) 1. X + XY = X (keep the boss) 2. XY + X'Z + YZ = XY + X'Z (consensus) a'b+ a'bc = a'b (X = a'b) a'bc' + bcd + a'bd = a'bc'+ bcd (X = c, Y = bd, Z = a'b)
6 Rule C  Eliminating Literals Rule D  Adding Redundant Terms X + X'Y = (X + X')(X + Y) = X + Y A'B + A'B'C'D' + ABCD' = A'(B + B'C'D') + ABCD' (common term: A') = A'(B + C'D') + ABCD' (Rule C) = B(A' + ACD') + A'C'D' (common term: B) = B(A' + CD') + A'C'D' (Rule C) = A'B + BCD' + A'C'D' (final terms) 1. Y = Y + XX' 2. Y = Y(X + X') 3. XY + X'Z = XY + X'Z + YZ 4. X = X + XY Add redundancy to eliminate other terms WX + XY + X'Z' + WY'Z' = WX + XY + X'Z' + WY'Z' + WZ' (add WZ' by consensus thm) = WX + XY + X'Z' + WZ' (eliminate WY'Z' by WZ') = WX + XY + X'Z' (consensus again) Another Example 23 A'B'C'D' + A'BC'D' + A'BD + A'BC'D + ABCD + ACD' + B'CD' = (Apply rules A, B, C, D) = A'C'D' + A'BD + B'CD' + ABC 24 Proving Validity of an Equation No easy way to determine when a Boolean expression has a minimum # of terms or literals Systematic way will be discussed in Unit 5 & Unit 6
7 How to Determine if an Equation Valid? Example Construct a truth table (proof by cases) 2. Manipulate one side until it is identical to the other side 3. Reduce both sides independently to the same expression 4. Perform the same operation on both sides if the operation is reversible (Boolean algebra ordinary algebra) Complement is reversible Multiplication/division and addition/subtraction are not reversible x+y = x+z does not imply y=z ( x=1, y=0, z=1) xy = xz does not imply y=z ( x=0, y=0, z=1) Using 2. and 3., usually 1. Reduce both sides to (minimum) SOP or POS 2. Compare both sides 3. Try to add or delete terms by using theorems Show that A'BD' + BCD + ABC' + AB'D = BC'D' + AD + A'BC By the consensus theorem, A'BD' + BCD + ABC' + AB'D = A'BD' + BCD + ABC' + AB'D + A'BC + BC'D' + ABD = AD + A'BD' + BCD + ABC' + A'BC + BC'D' = AD + A'BC + BC'D' One More Example Homework for Unit Show that A'BC'D + (A' + BC)(A + C'D') + BC'D + A'BC' = ABCD + A'C'D' + ABD + ABCD' + BC'D 1. Reduce the left side: A'BC'D + (A' + BC)(A + C'D') + BC'D + A'BC' = (A' + BC)(A + C'D') + BC'D + A'BC' = A'C'D' + ABC + BC'D + A'BC' (multiplying out) = A'C'D' + ABC + BC'D (consensus) 2. Reduce the right side: ABCD + A'C'D' + ABD + ABCD' + BC'D = ABC + A'C'D' + ABD + BC'D = ABC + A'C'D' + BC'D (consensus) 3. Because both sides were independently reduced to the same expression, the original equation is valid. Problems 3.12 (a) 3.13 (a) 3.23 (a) Homework #1 covers Units 13 Due 10am March 16, 2010 Submit your solutions to ED518 by the deadline. Quiz #1: March 23, 2010
Unit 3 Boolean Algebra (Continued)
Unit 3 Boolean Algebra (Continued) 1. ExclusiveOR Operation 2. Consensus Theorem Department of Communication Engineering, NCTU 1 3.1 Multiplying Out and Factoring Expressions Department of Communication
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 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 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 informationKarnaugh 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 informationUnited States Naval Academy Electrical and Computer Engineering Department. EC262 Exam 1
United States Naval Academy Electrical and Computer Engineering Department EC262 Exam 29 September 2. Do a page check now. You should have pages (cover & questions). 2. Read all problems in their entirety.
More informationCM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra  Factorisation
CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra  Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of
More informationChapter 2: Boolean Algebra and Logic Gates. Boolean Algebra
The Universit Of Alabama in Huntsville Computer Science Chapter 2: Boolean Algebra and Logic Gates The Universit Of Alabama in Huntsville Computer Science Boolean Algebra The algebraic sstem usuall used
More informationAlgebraic Properties and Proofs
Algebraic Properties and Proofs Name You have solved algebraic equations for a couple years now, but now it is time to justify the steps you have practiced and now take without thinking and acting without
More informationSection 1. Finding Common Terms
Worksheet 2.1 Factors of Algebraic Expressions Section 1 Finding Common Terms In worksheet 1.2 we talked about factors of whole numbers. Remember, if a b = ab then a is a factor of ab and b is a factor
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 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 informationClick on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is
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 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 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 information1.3 Polynomials and Factoring
1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.
More informationDigital circuits make up all computers and computer systems. The operation of digital circuits is based on
Digital Logic Circuits Digital circuits make up all computers and computer systems. The operation of digital circuits is based on Boolean algebra, the mathematics of binary numbers. Boolean algebra is
More information~ EQUIVALENT FORMS ~
~ EQUIVALENT FORMS ~ Critical to understanding mathematics is the concept of equivalent forms. Equivalent forms are used throughout this course. Throughout mathematics one encounters equivalent forms of
More informationOperations with Algebraic Expressions: Multiplication of Polynomials
Operations with Algebraic Expressions: Multiplication of Polynomials The product of a monomial x monomial To multiply a monomial times a monomial, multiply the coefficients and add the on powers with the
More informationCDA 3200 Digital Systems. Instructor: Dr. Janusz Zalewski Developed by: Dr. Dahai Guo Spring 2012
CDA 3200 Digital Systems Instructor: Dr. Janusz Zalewski Developed by: Dr. Dahai Guo Spring 2012 Outline MultiLevel Gate Circuits NAND and NOR Gates Design of TwoLevel Circuits Using NAND and NOR Gates
More informationElementary Logic Gates
Elementary Logic Gates Name Symbol Inverter (NOT Gate) ND Gate OR Gate Truth Table Logic Equation = = = = = + C. E. Stroud Combinational Logic Design (/6) Other Elementary Logic Gates NND Gate NOR Gate
More informationA Systematic Approach to Factoring
A Systematic Approach to Factoring Step 1 Count the number of terms. (Remember****Knowing the number of terms will allow you to eliminate unnecessary tools.) Step 2 Is there a greatest common factor? Tool
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 informationUnique column combinations
Unique column combinations Arvid Heise Guest lecture in Data Profiling and Data Cleansing Prof. Dr. Felix Naumann Agenda 2 Introduction and problem statement Unique column combinations Exponential search
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationSect 6.1  Greatest Common Factor and Factoring by Grouping
Sect 6.1  Greatest Common Factor and Factoring by Grouping Our goal in this chapter is to solve nonlinear equations by breaking them down into a series of linear equations that we can solve. To do this,
More informationChapter 5. Rational Expressions
5.. Simplify Rational Expressions KYOTE Standards: CR ; CA 7 Chapter 5. Rational Expressions Definition. A rational expression is the quotient P Q of two polynomials P and Q in one or more variables, where
More informationGates & Boolean Algebra. Boolean Operators. Combinational Logic. Introduction
Introduction Gates & Boolean lgebra Boolean algebra: named after mathematician George Boole (85 864). 2valued algebra. digital circuit can have one of 2 values. Signal between and volt =, between 4 and
More informationSection 8.8. 1. The given line has equations. x = 3 + t(13 3) = 3 + 10t, y = 2 + t(3 + 2) = 2 + 5t, z = 7 + t( 8 7) = 7 15t.
. The given line has equations Section 8.8 x + t( ) + 0t, y + t( + ) + t, z 7 + t( 8 7) 7 t. The line meets the plane y 0 in the point (x, 0, z), where 0 + t, or t /. The corresponding values for x and
More informationHow to bet using different NairaBet Bet Combinations (Combo)
How to bet using different NairaBet Bet Combinations (Combo) SINGLES Singles consists of single bets. I.e. it will contain just a single selection of any sport. The bet slip of a singles will look like
More informationBEGINNING ALGEBRA ACKNOWLEDMENTS
BEGINNING ALGEBRA The Nursing Department of Labouré College requested the Department of Academic Planning and Support Services to help with mathematics preparatory materials for its Bachelor of Science
More informationOnline EFFECTIVE AS OF JANUARY 2013
2013 A and C Session Start Dates (AB Quarter Sequence*) 2013 B and D Session Start Dates (BA Quarter Sequence*) Quarter 5 2012 1205A&C Begins November 5, 2012 1205A Ends December 9, 2012 Session Break
More information6 Commutators and the derived series. [x,y] = xyx 1 y 1.
6 Commutators and the derived series Definition. Let G be a group, and let x,y G. The commutator of x and y is [x,y] = xyx 1 y 1. Note that [x,y] = e if and only if xy = yx (since x 1 y 1 = (yx) 1 ). Proposition
More informationFINDING THE LEAST COMMON DENOMINATOR
0 (7 18) Chapter 7 Rational Expressions GETTING MORE INVOLVED 7. Discussion. Evaluate each expression. a) Onehalf of 1 b) Onethird of c) Onehalf of x d) Onehalf of x 7. Exploration. Let R 6 x x 0 x
More informationCOMMUTATIVE RINGS. Definition: A domain is a commutative ring R that satisfies the cancellation law for multiplication:
COMMUTATIVE RINGS Definition: A commutative ring R is a set with two operations, addition and multiplication, such that: (i) R is an abelian group under addition; (ii) ab = ba for all a, b R (commutative
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 informationHow To Prove The Triangle Angle Of A Triangle
Simple trigonometric substitutions with broad results Vardan Verdiyan, Daniel Campos Salas Often, the key to solve some intricate algebraic inequality is to simplify it by employing a trigonometric substitution.
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More information1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes
Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.
More informationGeo, Chap 4 Practice Test, EV Ver 1
Class: Date: Geo, Chap 4 Practice Test, EV Ver 1 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. (43) In each pair of triangles, parts are congruent as
More informationIntermediate Math Circles October 10, 2012 Geometry I: Angles
Intermediate Math Circles October 10, 2012 Geometry I: Angles Over the next four weeks, we will look at several geometry topics. Some of the topics may be familiar to you while others, for most of you,
More informationGeometry Module 4 Unit 2 Practice Exam
Name: Class: Date: ID: A Geometry Module 4 Unit 2 Practice Exam Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which diagram shows the most useful positioning
More informationwww.mohandesyar.com SOLUTIONS MANUAL DIGITAL DESIGN FOURTH EDITION M. MORRIS MANO California State University, Los Angeles MICHAEL D.
27 Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This publication is protected by opyright and written permission should be obtained or likewise. For information regarding permission(s),
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 16, 2012 8:30 to 11:30 a.m.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 16, 2012 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of your
More informationData Mining: Partially from: Introduction to Data Mining by Tan, Steinbach, Kumar
Data Mining: Association Analysis Partially from: Introduction to Data Mining by Tan, Steinbach, Kumar Association Rule Mining Given a set of transactions, find rules that will predict the occurrence of
More informationLecture Notes on Database Normalization
Lecture Notes on Database Normalization Chengkai Li Department of Computer Science and Engineering The University of Texas at Arlington April 15, 2012 I decided to write this document, because many students
More informationData Mining Apriori Algorithm
10 Data Mining Apriori Algorithm Apriori principle Frequent itemsets generation Association rules generation Section 6 of course book TNM033: Introduction to Data Mining 1 Association Rule Mining (ARM)
More information1.4. Removing Brackets. Introduction. Prerequisites. Learning Outcomes. Learning Style
Removing Brackets 1. Introduction In order to simplify an expression which contains brackets it is often necessary to rewrite the expression in an equivalent form but without any brackets. This process
More informationClass One: Degree Sequences
Class One: Degree Sequences For our purposes a graph is a just a bunch of points, called vertices, together with lines or curves, called edges, joining certain pairs of vertices. Three small examples of
More informationBenchmark Databases for Testing BigData Analytics In Cloud Environments
North Carolina State University Graduate Program in Operations Research Benchmark Databases for Testing BigData Analytics In Cloud Environments Rong Huang Rada Chirkova Yahya Fathi ICA CON 2012 April
More informationGeometry Regents Review
Name: Class: Date: Geometry Regents Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. If MNP VWX and PM is the shortest side of MNP, what is the shortest
More informationAngles in a Circle and Cyclic Quadrilateral
130 Mathematics 19 Angles in a Circle and Cyclic Quadrilateral 19.1 INTRODUCTION You must have measured the angles between two straight lines, let us now study the angles made by arcs and chords in a circle
More informationSolutions Manual for How to Read and Do Proofs
Solutions Manual for How to Read and Do Proofs An Introduction to Mathematical Thought Processes Sixth Edition Daniel Solow Department of Operations Weatherhead School of Management Case Western Reserve
More information2 : two cube. 5 : five cube. 10 : ten cube.
Math 105 TOPICS IN MATHEMATICS REVIEW OF LECTURES VI Instructor: Line #: 52920 Yasuyuki Kachi 6 Cubes February 2 Mon, 2015 We can similarly define the notion of cubes/cubing Like we did last time, 3 2
More informationHow To Solve Factoring Problems
05W4801AM1.qxd 8/19/08 8:45 PM Page 241 Factoring, Solving Equations, and Problem Solving 5 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
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 informationA single register, called the accumulator, stores the. operand before the operation, and stores the result. Add y # add y from memory to the acc
Other architectures Example. Accumulatorbased machines A single register, called the accumulator, stores the operand before the operation, and stores the result after the operation. Load x # into acc
More informationLinear Equations in One Variable
Linear Equations in One Variable MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this section we will learn how to: Recognize and combine like terms. Solve
More informationCombinational circuits
Combinational circuits Combinational circuits are stateless The outputs are functions only of the inputs Inputs Combinational circuit Outputs 3 Thursday, September 2, 3 Enabler Circuit (Highlevel view)
More informationFunctional Dependencies and Normalization
Functional Dependencies and Normalization 5DV119 Introduction to Database Management Umeå University Department of Computing Science Stephen J. Hegner hegner@cs.umu.se http://www.cs.umu.se/~hegner Functional
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, January 26, 2016 1:15 to 4:15 p.m., only.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, January 26, 2016 1:15 to 4:15 p.m., only Student Name: School Name: The possession or use of any communications
More informationHOW TO USE MINITAB: DESIGN OF EXPERIMENTS. Noelle M. Richard 08/27/14
HOW TO USE MINITAB: DESIGN OF EXPERIMENTS 1 Noelle M. Richard 08/27/14 CONTENTS 1. Terminology 2. Factorial Designs When to Use? (preliminary experiments) Full Factorial Design General Full Factorial Design
More informationTheory behind Normalization & DB Design. Satisfiability: Does an FD hold? Lecture 12
Theory behind Normalization & DB Design Lecture 12 Satisfiability: Does an FD hold? Satisfiability of FDs Given: FD X Y and relation R Output: Does R satisfy X Y? Algorithm: a.sort R on X b.do all the
More informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
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. The QuineMcCluskey Method Handout 5 January 21, 2016. CSEE E6861y Prof. Steven Nowick
CSEE E6861y Prof. Steven Nowick The QuineMcCluskey Method Handout 5 January 21, 2016 Introduction The QuineMcCluskey method is an exact algorithm which finds a minimumcost sumofproducts implementation
More informationMathematics 3301001 Spring 2015 Dr. Alexandra Shlapentokh Guide #3
Mathematics 3301001 Spring 2015 Dr. Alexandra Shlapentokh Guide #3 The problems in bold are the problems for Test #3. As before, you are allowed to use statements above and all postulates in the proofs
More informationUnderstanding Logic Design
Understanding Logic Design ppendix of your Textbook does not have the needed background information. This document supplements it. When you write add DD R0, R1, R2, you imagine something like this: R1
More information6.5 Factoring Special Forms
440 CHAPTER 6. FACTORING 6.5 Factoring Special Forms In this section we revisit two special product forms that we learned in Chapter 5, the first of which was squaring a binomial. Squaring a binomial.
More informationFactoring (pp. 1 of 4)
Factoring (pp. 1 of 4) Algebra Review Try these items from middle school math. A) What numbers are the factors of 4? B) Write down the prime factorization of 7. C) 6 Simplify 48 using the greatest common
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 informationLogic Reference Guide
Logic eference Guide Advanced Micro evices INTOUCTION Throughout this data book and design guide we have assumed that you have a good working knowledge of logic. Unfortunately, there always comes a time
More informationDigital Logic Design. Basics Combinational Circuits Sequential Circuits. PuJen Cheng
Digital Logic Design Basics Combinational Circuits Sequential Circuits PuJen Cheng Adapted from the slides prepared by S. Dandamudi for the book, Fundamentals of Computer Organization and Design. Introduction
More informationTo Evaluate an Algebraic Expression
1.5 Evaluating Algebraic Expressions 1.5 OBJECTIVES 1. Evaluate algebraic expressions given any signed number value for the variables 2. Use a calculator to evaluate algebraic expressions 3. Find the sum
More informationCHAPTER 8 QUADRILATERALS. 8.1 Introduction
CHAPTER 8 QUADRILATERALS 8.1 Introduction You have studied many properties of a triangle in Chapters 6 and 7 and you know that on joining three noncollinear points in pairs, the figure so obtained is
More information3.1 Triangles, Congruence Relations, SAS Hypothesis
Chapter 3 Foundations of Geometry 2 3.1 Triangles, Congruence Relations, SAS Hypothesis Definition 3.1 A triangle is the union of three segments ( called its side), whose end points (called its vertices)
More informationFinding the Measure of Segments Examples
Finding the Measure of Segments Examples 1. In geometry, the distance between two points is used to define the measure of a segment. Segments can be defined by using the idea of betweenness. In the figure
More informationHow To Factor By Gcf In Algebra 1.5
72 Factoring by GCF Warm Up Lesson Presentation Lesson Quiz Algebra 1 Warm Up Simplify. 1. 2(w + 1) 2. 3x(x 2 4) 2w + 2 3x 3 12x Find the GCF of each pair of monomials. 3. 4h 2 and 6h 2h 4. 13p and 26p
More informationWarmup Tangent circles Angles inside circles Power of a point. Geometry. Circles. Misha Lavrov. ARML Practice 12/08/2013
Circles ARML Practice 12/08/2013 Solutions Warmup problems 1 A circular arc with radius 1 inch is rocking back and forth on a flat table. Describe the path traced out by the tip. 2 A circle of radius
More informationYou must have: Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser, calculator. Tracing paper may be used.
Write your name here Surname Other names Edexcel IGCSE Mathematics B Paper 1 Centre Number Candidate Number Monday 6 June 2011 Afternoon Time: 1 hour 30 minutes Paper Reference 4MB0/01 You must have: Ruler
More informationRelational Database Design
Relational Database Design To generate a set of relation schemas that allows  to store information without unnecessary redundancy  to retrieve desired information easily Approach  design schema in appropriate
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationChapter 3. Inversion and Applications to Ptolemy and Euler
Chapter 3. Inversion and Applications to Ptolemy and Euler 2 Power of a point with respect to a circle Let A be a point and C a circle (Figure 1). If A is outside C and T is a point of contact of a tangent
More information#6 Opener Solutions. Move one more spot to your right. Introduce yourself if needed.
1. Sit anywhere in the concentric circles. Do not move the desks. 2. Take out chapter 6, HW/notes #1#5, a pencil, a red pen, and your calculator. 3. Work on opener #6 with the person sitting across from
More informationHow To Find Out What A Key Is In A Database Engine
Database design theory, Part I Functional dependencies Introduction As we saw in the last segment, designing a good database is a non trivial matter. The E/R model gives a useful rapid prototyping tool,
More informationDie Welt MultimediaReichweite
Die Welt MultimediaReichweite 1) Background The quantification of Die Welt s average daily audience (known as MultimediaReichweite, MMR) has been developed by Die Welt management, including the research
More informationexclusiveor and Binary Adder R eouven Elbaz reouven@uwaterloo.ca Office room: DC3576
exclusiveor and Binary Adder R eouven Elbaz reouven@uwaterloo.ca Office room: DC3576 Outline exclusive OR gate (XOR) Definition Properties Examples of Applications Odd Function Parity Generation and Checking
More information5.1 Midsegment Theorem and Coordinate Proof
5.1 Midsegment Theorem and Coordinate Proof Obj.: Use properties of midsegments and write coordinate proofs. Key Vocabulary Midsegment of a triangle  A midsegment of a triangle is a segment that connects
More informationVisa Smart Debit/Credit Certificate Authority Public Keys
CHIP AND NEW TECHNOLOGIES Visa Smart Debit/Credit Certificate Authority Public Keys Overview The EMV standard calls for the use of Public Key technology for offline authentication, for aspects of online
More informationAlgebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.
Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear
More informationBlue Pelican Geometry Theorem Proofs
Blue Pelican Geometry Theorem Proofs Copyright 2013 by Charles E. Cook; Refugio, Tx (All rights reserved) Table of contents Geometry Theorem Proofs The theorems listed here are but a few of the total in
More informationName Intro to Algebra 2. Unit 1: Polynomials and Factoring
Name Intro to Algebra 2 Unit 1: Polynomials and Factoring Date Page Topic Homework 9/3 2 Polynomial Vocabulary No Homework 9/4 x In Class assignment None 9/5 3 Adding and Subtracting Polynomials Pg. 332
More informationCIRCLE THEOREMS. Edexcel GCSE Mathematics (Linear) 1MA0
Edexcel GCSE Mathematics (Linear) 1MA0 CIRCLE THEOREMS Materials required for examination Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser. Tracing paper may
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any
More informationNegative Integer Exponents
7.7 Negative Integer Exponents 7.7 OBJECTIVES. Define the zero exponent 2. Use the definition of a negative exponent to simplify an expression 3. Use the properties of exponents to simplify expressions
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationScilab Textbook Companion for Digital Electronics: An Introduction To Theory And Practice by W. H. Gothmann 1
Scilab Textbook Companion for Digital Electronics: An Introduction To Theory And Practice by W. H. Gothmann 1 Created by Aritra Ray B.Tech Electronics Engineering NITDURGAPUR College Teacher Prof. Sabyasachi
More information