CH3 Boolean Algebra (cont d)


 Naomi Stanley
 1 years ago
 Views:
Transcription
1 CH3 Boolean Algebra (cont d) Lecturer: 吳 安 宇 Date:2005/10/7 ACCESS IC LAB
2 v Today, you ll know: Introduction 1. Guidelines for multiplying out/factoring expressions 2. ExclusiveOR and Equivalence operations 3. Positive logic and negative logic 4. More about consensus theorem 5. Algebraic simplification of switching expressions 6. Approach to prove validity of an equation 7. The difference between ordinary algebra and Boolean algebra pp. 2
3 Guidelines for Multiplying Out and Factoring vuse X(Y+Z) = XY + XZ...(1) (X+Y)(X+Z) = X + YZ...(2) (X+Y)(X +Z) = XZ + X Y...(3) vfor multiplying out, (2) and (3) should be generally applied before (1) to avoid generating unnecessary terms vfor factoring, apply (1), (2), (3) from right terms to left terms pp. 3
4 Multiplying Out Expression EX. F = (Q + AB)(C D + Q ) = QC D + Q AB or F = QC D + QQ + AB C D + AB Q EX. (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)] Distributed Law = (A+B+C DE)(AC+A D +A E) = AC+ABC+A BD +A BE+A C DE (SOP form) => By brute force => 162 terms pp. 4
5 Factoring Expression v EX. AC + A BD + A BE + A C DE = AC + A (BD + BE + C DE) XZ + X Y = (X + Y)(X + Z) = (A + BD + BE + C DE)(A + C) = [ A + C DE + B (D + E) ](A + C) X + YZ = (X+Y)(X+Z) = (A + C DE + B)(A + C DE + D + E)(A + C) = (A + B + C )(A + B + D)(A + B + E)(A + D + E)(A + C) pp. 5
6 3.2 ExclusiveOR Operations vexclusiveor (XOR) X Y X Y Truth Table Symbol Boolean Expression : X Y = X Y + XY pp. 6
7 ExclusiveOR Operations vuseful Theorems : X 0 = X X Y = Y X (commutative) X 1 = X (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 = XY + X Y pp. 7
8 Proof of Distributive Laws vxy XZ = XY(XZ) + (XY) XZ = XY(X + Z ) + (X + Y )XZ = XYZ +XY Z = X(YZ + Y Z) = X(Y Z) pp. 8
9 Equivalence Operations (Exclusive NOR) X Y X Y (X Y) X Y = XY + X Y pp. 9
10 Simplification of XOR and XNOR vx X Y = X Y + XY Y = X Y + XY EX (see p.62). F = (A B C) + (B AC ) = [(A B)C + (A B) C ] + [B (AC ) + B(AC ) ] = A BC + (A+B )C + AB C + B(A +C) = B(A C + A + C) + C(A + B + AB ) = B(A + C) + C (A + B ) ( can be further simplified) pp. 10
11 3.3 Consensus Theorem 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 pp. 11
12 How to Find Consensus Term? 1. Find a pair of terms, one of which contains a variable and the other contains 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) BD BC = BCD (redundant term) pp. 12
13 Consensus Theorem vapplication to eliminate redundant terms from Boolean Expressions a b + ac + bc + b c +ab = a b + ac + bc pp. 13
14 Consensus Theorem Dual form of consensus theorem (X + Y)(X + Z)(Y + Z) = (X + Y)(X + Z) Example (others are on p.67) : (a + b + c )(a + b + d )(b + c + d ) = (a + b + c )(b + c + d ) (a+ b + c ) + (b + c +d ) a+b + b+d = a+b+d Simplification of Boolean Expression can reduce the cost of realizing the network using gates pp. 14
15 Algebraic Simplification of Switching Expression va. Combining Terms XY + XY =X(Y + Y ) = X EX.1 abc d + abcd = abd (X = abd, Y = c) EX.2 ab c + abc + a bc = ab c + abc + abc + a bc = ac + bc EX.3 (a + bc)(d + e ) + a (b + c )(d + e ) = d + e pp. 15
16 Algebraic Simplification of Switching Expression vrule B  Eliminating Terms : X + XY = X XY + X Z + YZ = XY + X Z EX.1 a b + a bc = a b (X = a b) a bc + bcd + a bd = a bc + bcd (X = c, Y = bd, Z = a b) pp. 16
17 Algebraic Simplification of Switching Expression vrule C  Eliminating Literals : X + X Y = (X + X )(X + Y) = X + Y EX. 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) pp. 17
18 Algebraic Simplification of Switching Expression vrule D  Adding Redundant Terms vadd XX = 0 vmultiply by (X + X ) = 1 vadd YZ to (XY + X Z) (reverse of Consensus) Because XY + X Z + YZ = XY + X Z vadd XY to X pp. 18
19 Algebraic Simplification of Switching Expression vex.1 of Adding Redundant Terms WX + XY + X Z + WY Z = WX + XY + X Z + WY Z + W Z (add W Z by Consensus Theorem) = WX + XY + X Z + WZ (eliminate WY Z by WZ ) = WX + XY + X Z pp. 19
20 Algebraic Simplification of Switching Expression EX.2 A B C D + A BC D + A BD + A BC D + ABCD + ACD + B CD = A C D + A BD + B CD + ABC (A, B, C, D methods are applied) No easy way to determine when a Boolean Expression has a min. no. of terms or literals Systematic way is presented in Ch.5 & CH.6 pp. 20
21 Proving Validity of an Equation vapproach : vconstruct a Truth Table vmanipulate one side of the equation till it s identical to the other side vreduce both sides independently to the same equation v(a) Perform same operation on both sides (b) Cannot Subtract or Divide both sides (Subtraction, Division NOT defined) pp. 21
22 Proving Validity of an Equation vusually : vreduce both sides to Sum of Products (SOP) vcompare both sides vtry to Add or Delete terms by using Theorems pp. 22
23 Proving Validity of an Equation vex.1 Show that A BD + BCD + ABC + AB D = BC D + AD + A BC By Consensus Theorem : A BD + BCD + ABC + AB D + BC D + A BC + ABD = AD + A BD + BCD + ABC + BC D + A BC = AD + BC D + A BC pp. 23
24 Proving Validity of an Equation vex.2 Show A BC D + (A + BC)(A + C D ) + BC D + A BC = ABCD + A C D + ABD + ABCD + BC D Reducing the left side A BC D + (A + BC)(A + C D ) + BC D + A BC = (A + BC)(A + C D ) + BC D + A BC = ABC + A C D + BC D + A BC = ABC + A C D + BC D pp. 24
25 Proving Validity of an Equation vex.2(cont.) vreducing the left side ABCD + A C D + ABD + ABCD + BC D = ABC + A C D + ABD + BC D = ABC + A C D + BC D Because both sides were independently reduced to the same expression, the original equation is valid pp. 25
26 Boolean Algebra & Ordinary Algebra vboolean Algebra Ordinary Algebra EX.1 X + Y = X + Z => Y = Z (?) X = 1, Y = 0 => = But 0 1 EX.2 If XY = XZ then Y = Z True : when X 0 False : when X = 0 pp. 26
27 Boolean Algebra & Ordinary Algebra vex.3 if Y = Z then X + Y = X + Z (V) if Y = Z then XY = XZ (V) Add/Multiply the same term => Valid Subtract/Divide the same term => Not Valid Check programmed exercise 3.1, 3.2,,3.5 for practice pp. 27
Boolean 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 informationUnit 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 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 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 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 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 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 informationENEE 244 (01**). Spring 2006. Homework 4. Due back in class on Friday, April 7.
ENEE 244 (**). Spring 26 Homework 4 Due back in class on Friday, April 7.. Implement the following Boolean expression with exclusiveor and AND gates only: F = AB'CD' + A'BCD' + AB'C'D + A'BC'D. F = AB
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 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 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 informationAn Interesting Way to Combine Numbers
An Interesting Way to Combine Numbers Joshua Zucker and Tom Davis November 28, 2007 Abstract This exercise can be used for middle school students and older. The original problem seems almost impossibly
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 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 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 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 informationEquivalence Relations
Equivalence Relations Definition An equivalence relation on a set S, is a relation on S which is reflexive, symmetric and transitive. Examples: Let S = Z and define R = {(x,y) x and y have the same parity}
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 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 information1.4 Variable Expressions
1.4 Variable Expressions Now that we can properly deal with all of our numbers and numbering systems, we need to turn our attention to actual algebra. Algebra consists of dealing with unknown values. These
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 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 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 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 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 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 informationEXPERIMENT NO.1:INTRODUCTION TO BASIC GATES AND LOGIC SIMPLIFICATION TECHNIQUES
DEPARTMENT OF ELECTRICAL AND ELECTROINC ENGINEERING BANGLADESH UNIVERSITY OF ENGINEERING & TECHNOLOGY EEE 304 : Digital Electronics Laboratory EXPERIMENT NO.1:INTRODUCTION TO BASIC GATES AND LOGIC SIMPLIFICATION
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 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 informationELEC2200 Digital Circuits and Systems Fall 2016 Instructor: Levent Yobas
Lecture 3b 1 ELEC2200 Digital Circuits and Systems Fall 2016 Instructor: Levent Yobas Lecture 3b Gate Level Implementation Lecture 3b 2 Lecture Overview Implementations Using ANDOR, OR AND Using NANDNAND,
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 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 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 informationLESSON 6.2 POLYNOMIAL OPERATIONS I
LESSON 6.2 POLYNOMIAL OPERATIONS I Overview In business, people use algebra everyday to find unknown quantities. For example, a manufacturer may use algebra to determine a product s selling price in order
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 informationBasics of Digital Systems. Boolean algebra Truth tables Karnaugh maps
Basics of Digital Systems Boolean algebra Truth tables Karnaugh maps 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
More informationNOT AND OR XOR NAND NOR
NOT AND OR XOR NAND NOR Expression 1: It is raining today Expression 2: Today is my birthday X Meaning True False It is raining today It is not raining Binary representation of the above: X Meaning 1 It
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 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 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 informationUNIT  II LOGIC GATES AND GATES CLASSIFICATION
UNIT  II Logic Gates: Gates Classifications: Basic Gates (AND, OR, NOT), Universal Gates (NAND, NOR), Exclusive Gates (XOR, XNOR)(except circuit diagram) Logic Symbols, Logic Operators, Logical expression
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 informationArithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get
Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real
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 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 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 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 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 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 informationGates, Circuits and Boolean Functions
Lecture 2 Gates, Circuits and Boolean Functions DOC 112: Hardware Lecture 2 Slide 1 In this lecture we will: Introduce an electronic representation of Boolean operators called digital gates. Define a schematic
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 informationSchema Refinement & Normalization Theory
Schema Refinement & Normalization Theory Functional Dependencies Week 111 1 What s the Problem Consider relation obtained (call it SNLRHW) Hourly_Emps(ssn, name, lot, rating, hrly_wage, hrs_worked) What
More informationTwice the Angle.  Circle Theorems 3: Angle at the Centre Theorem 
 Circle Theorems 3: Angle at the Centre Theorem  Definitions An arc of a circle is a contiguous (i.e. no gaps) portion of the circumference. An arc which is half of a circle is called a semicircle.
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 informationPRIMARY CONTENT MODULE Algebra I Linear Equations & Inequalities T71. Applications. F = mc + b.
PRIMARY CONTENT MODULE Algebra I Linear Equations & Inequalities T71 Applications The formula y = mx + b sometimes appears with different symbols. For example, instead of x, we could use the letter C.
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 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 informationSimilar Polygons. Copy both triangles onto tracing paper. Measure and record the sides of each triangle. Cut out both triangles.
7 Similar Polygons MAIN IDEA Identify similar polygons and find missing measures of similar polygons. New Vocabulary polygon similar corresponding parts congruent scale factor Math Online glencoe.com
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 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 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 get the Simplified Expanded Form of a polynomial, II
How to get the Simplified Expanded Form of a polynomial, II Nikos Apostolakis September 29, 2010 Recall. Recall that the distributive law states that multiplication distributes over addition and subtraction:
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 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 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 informationGeometry Chapter 7. Ratios & Proportions Properties of Proportions Similar Polygons Similarity Proofs Triangle Angle Bisector Theorem
Geometry Chapter 7 Ratios & Proportions Properties of Proportions Similar Polygons Similarity Proofs Triangle Angle Bisector Theorem Name: Geometry Assignments Chapter 7 Date Due Similar Polygons Section
More informationFactoring. Factoring Monomials Monomials can often be factored in more than one way.
Factoring Factoring is the reverse of multiplying. When we multiplied monomials or polynomials together, we got a new monomial or a string of monomials that were added (or subtracted) together. For example,
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 informationFactoring Quadratic Expressions
Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the
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 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 informationProperties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
More informationDefinition of an nth Root
Radicals and Complex Numbers 7 7. Definition of an nth Root 7.2 Rational Exponents 7.3 Simplifying Radical Expressions 7.4 Addition and Subtraction of Radicals 7.5 Multiplication of Radicals 7.6 Rationalization
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 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 information2011, The McGrawHill Companies, Inc. Chapter 4
Chapter 4 4.1 The Binary Concept Binary refers to the idea that many things can be thought of as existing in only one of two states. The binary states are 1 and 0 The 1 and 0 can represent: ON or OFF Open
More informationMODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.
MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on
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 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 informationFactoring, Solving. Equations, and Problem Solving REVISED PAGES
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 informationMth 95 Module 2 Spring 2014
Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression
More informationChapter Two. Deductive Reasoning
Chapter Two Deductive Reasoning Objectives A. Use the terms defined in the chapter correctly. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply
More informationSPECIAL PRODUCTS AND FACTORS
CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 111 Factors and Factoring 112 Common Monomial Factors 113 The Square of a Monomial 114 Multiplying the Sum and the Difference of Two Terms 115 Factoring the
More informationGeometry CP Lesson 51: Bisectors, Medians and Altitudes Page 1 of 3
Geometry CP Lesson 51: Bisectors, Medians and Altitudes Page 1 of 3 Main ideas: Identify and use perpendicular bisectors and angle bisectors in triangles. Standard: 12.0 A perpendicular bisector of a
More informationLab Manual. Digital System Design (Pr): COT215 Digital Electronics (P): IT211
Lab Manual Digital System Design (Pr): COT215 Digital Electronics (P): IT211 Lab Instructions Several practicals / programs? Whether an experiment contains one or several practicals /programs One practical
More informationSimple trigonometric substitutions with broad results
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 informationRULE 1: Additive Identity Property
RULE 1: Additive Identity Property Additive Identity Property a + 0 = a x + 0 = x If we add 0 to any number, we will end up with the same number. Zero is represented through the the green vortex. When
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 informationChapter 4. Polynomials
4.1. Add and Subtract Polynomials KYOTE Standards: CR 8; CA 2 Chapter 4. Polynomials Polynomials in one variable are algebraic expressions such as 3x 2 7x 4. In this example, the polynomial consists of
More informationDigital Logic cct. Lec. (6)
THE NAND GATE The NAND gate is a popular logic element because it can be used as a universal gate: that is, NAND gates can be used in combination to perform the AND, OR, and inverter operations. The term
More informationexpression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.
A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are
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 informationUsing the ac Method to Factor
4.6 Using the ac Method to Factor 4.6 OBJECTIVES 1. Use the ac test to determine factorability 2. Use the results of the ac test 3. Completely factor a trinomial In Sections 4.2 and 4.3 we used the trialanderror
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 informationYear 10 Term 1 Homework
Yimin Math Centre Year 10 Term 1 Homework Student Name: Grade: Date: Score: Table of contents 10 Year 10 Term 1 Week 10 Homework 1 10.1 Deductive geometry.................................... 1 10.1.1 Basic
More informationSample Problems. Lecture Notes Similar Triangles page 1
Lecture Notes Similar Triangles page 1 Sample Problems 1. The triangles shown below are similar. Find the exact values of a and b shown on the picture below. 2. Consider the picture shown below. a) Use
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 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 information