Chapter 2: Boolean Algebra and Logic Gates. Boolean Algebra

Size: px
Start display at page:

Download "Chapter 2: Boolean Algebra and Logic Gates. Boolean Algebra"

Transcription

1 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 to work with binar logic epressions Postulates: 1. Closure: An defined operation on (0, 1) gives (0,1) 2. Identit: 0 + = ; 1. = 3. Commutative: + = + ; = 4. Distributive: ( + ) = + ; + () = ( + )( + ) 5. Def of Complement: + = 1; = 0 6. At least 2 elements (0 and 1) Precedence rule: (1) parentheses (2) NOT (3) AND (4) OR 1

2 The Universit Of Alabama in Huntsville Computer Science The Dualit Principle A Boolean epression that is alwas true is still true if we echange OR with AND and 0 with 1 Eamples: + = 1 so: = 0 + = + so: = Note that we cannot use Dualit to sa that + =1, so = 0 Wh not? The Universit Of Alabama in Huntsville Computer Science Useful Postulates and Theorems (a) Postulate =. 1 = Postulate 5 + = 1 = 0 Theorem 1 + = = Theorem = 1. 0 = 0 Theorem 3 (involution) ( ) = Postulate 3 (commutative) + = + = Theorem 4 (associative) + ( + ) = ( + ) = () = () Postulate 4 (distributive) ( + ) = + + = ( + )( + ) Theorem 5 (demorgan s Law) ( + ) = () = + Theorem 6 (absorption) + = ( + ) = (b) 2

3 The Universit Of Alabama in Huntsville Computer Science Eample: Theorem 1: + = ; = Proving the Theorems Proof: + = ( + ) 1 postulate 2(b) = ( + )( + ) 5(a) = + 4(b) = + 0 5(b) = 2(a) = b dualit The Universit Of Alabama in Huntsville Computer Science Proving b Truth Table Two Boolean epressions are equal in all cases if and onl if the have the same Truth Table. (You ma use this to prove the epressions are equal unless I sa otherwise). Eample: Prove demorgan s Law: ( + ) = ( + ) ( + ) The Truth Table of ( + ) is equal to the Truth Table of, so we know that ( + ) = for all values of and. 3

4 The Universit Of Alabama in Huntsville Computer Science Boolean functions and circuit equivalents The Universit Of Alabama in Huntsville Computer Science Implementing a Boolean epression as a circuit F1 = + F1 F1 F1 4

5 The Universit Of Alabama in Huntsville Computer Science Simplifing epressions There are man different was to write the same epression Eample: + + = + Different forms of the epression will require different numbers of gates to implement Proof? See page 45 in tet Generall, longer epressions with more terms require more gates and/or more comple gates More gates higher power, higher cost, larger sie, So finding a wa to simplif epressions will pa off in terms of the circuits we design The Universit Of Alabama in Huntsville Computer Science A metric for use in simplifing epressions Define a literal as each occurrence of a variable in the epression Eample: F2 = literals If we can write the epression with fewer literal, we will consider it to be simpler (and to take fewer gates). Note that this is a rule of thumb and does not alwas give an optimum answer 5

6 The Universit Of Alabama in Huntsville Computer Science Simplifing epressions using the postulates and theorems of Boolean Algebra From page of tet 1. ( + ) (3 literals) = + p4a = 0 + p5b = p2a (2 literals) 2. + = + The dual of (1) 3. ( + )( + ) (4 literals) = + p4b = + 0 p5b = p2a (1 literal) (6 literals) = + + (1) p2b = + + ( + ) p5a = p4a = p3b twice = p3a twice = p2b twice = (1 + ) + (1 + ) p4a twice = T2a twice = + p2a twice (4 literals) 5. (+)( +)(+) = (+)( +) The dual of (4) The Consensus Theorem The Universit Of Alabama in Huntsville Computer Science Complementing a function demorgan s Law sas: ( + ) = To take (A + B + C) Let = B+C Then (A + B + C) = (A + ) = A = A (B + C) = A B C In general: (A+B+C+D+ ) = A B C D ; (ABCD ) = A +B +C +D A more comple function: F = + F = ( + ) = ( ) ( ) = ( + + )( + + ) 6

7 The Universit Of Alabama in Huntsville Computer Science A shortcut for complementing a function To complement a function, ou can take the dual of the function, and complement each literal. For the previous eample: F = + dual of F = ( + + )( + + ) so F = ( + + )( + + ) The Universit Of Alabama in Huntsville Computer Science Standard forms of Boolean Epressions 7

8 The Universit Of Alabama in Huntsville Computer Science Definitions Product term a term consisting of literals ANDed together» Eample: AB F Minterm a Product term in which all variables appear» Eample: ABC D where A,B,C, and D are the variables of the function Sum term a term consisting of literals ORed together» Eample: A + B + F Materm a Sum term in which all variables appear» Eample: A + B + C + D where A, B, C, and D are the variables of the function The Universit Of Alabama in Huntsville Computer Science SOP and Canonical SOP Form A function is in Sum of Products (SOP) form if it is written as product terms ORed together Eample: f( ) = + + A function is in Canonical SOP form if it is in SOP form and all terms are minterms Eample: g( ) = + + 8

9 The Universit Of Alabama in Huntsville Computer Science POS and Canonical POS form A function is in Product of Sums (POS) form if it is written as sum terms ANDed together Eample: f( ) = ( + + ) ( + ) () A function is in Canonical POS form if it is written in POS form and all terms are Materms Eample: g( ) = ( + + ) ( + + ) The Universit Of Alabama in Huntsville Computer Science Minterms and the Truth Table Each row of a Truth Table corresponds to a minterm f( ) minterm m m m m m m m m 7 f( ) = Minterm List Form: f( ) = Σm(1, 4, 5, 7) The 1 s of the Truth Table show the minterms that are in the Canonical SOP epression 9

10 The Universit Of Alabama in Huntsville Computer Science f() f( ) = + + = Σm(1,4,7) Eamples A B C D g(abcd) g(a B C D) = A B C D + AB CD + ABC D = Σm(1, 10, 13) The Universit Of Alabama in Huntsville Computer Science Materms and the Truth Table Each row of a Truth Table corresponds to a materm f( ) Materm M M M M M M M M f( ) = (++)(+ +)( + + ) Materm List Form: f( ) = ΠM(0,3,6) Note the differences from the wa minterms are complemented The 0 s of the Truth Table show the materms that are in the Canonical POS epression 10

11 The Universit Of Alabama in Huntsville Computer Science f() Eample f( ) = (++)(+ +)(+ + )( ++ )( + +) = ΠM(0, 2, 3, 5, 6) Note that the Minterm List and Materm List taken together include the number of ever row of the Truth Table. That means that if ou determine either one of the lists, ou can determine the other one b simpl writing the row numbers that are not in the first one. Eamples: If F(ABC) = Σm(0-3), then F(ABC) = ΠM(4-7) if G(w) = ΠM(0,12,15), then G(w) = Σm(1-11, 13, 14) The Universit Of Alabama in Huntsville Computer Science Basic Combinational Circuit Designs 11

12 The Universit Of Alabama in Huntsville Computer Science SOP to AND-OR An SOP epression can be directl implemented in a two-level combinational circuit with an AND gate for each product term and an OR gate to combine the terms Eample: f() = + + f() The Universit Of Alabama in Huntsville Computer Science POS to OR-AND A POS epression can be directl implemented in a two-level combinational circuit with an OR gate for each sum term and an AND gate to combine the terms Eample: f(w) = (+)(w+) w f(w) 12

13 The Universit Of Alabama in Huntsville Computer Science Circuits for mied-form epressions Combinational circuits for mied-form epressions ma have more than two levels Eample: f(abcde) = AB + C(D + E) A B C D E f(abcde) The Universit Of Alabama in Huntsville Computer Science NAND NAND Other common gate tpes () = + Eclusive-OR (XOR) XOR = + NOR NOR (+) = 13

14 The Universit Of Alabama in Huntsville Computer Science You can use NAND and NOR to do anthing ou can do with AND, OR, and NOT () = (() ) = ( ) = + NOT AND OR (+) = ( + ) = = ((+) ) = + The Universit Of Alabama in Huntsville Computer Science SOP to NAND (1) We alread determined that we can go directl from SOP form to an AND-OR implementation f() = + + f() We can substitute the NAND equivalents for the AND and OR gates f() 14

15 The Universit Of Alabama in Huntsville Computer Science SOP to NAND (2) The circled gates are just 2 inverters in series the do nothing So leave them out f() f() The Universit Of Alabama in Huntsville Computer Science Check: Is this still the original f()? SOP to NAND (3) The circuit produces: ( ( ) ( )) = + ( ) + ( ) = + + f() 15

16 The Universit Of Alabama in Huntsville Computer Science SOP to NAND (4) We can use NAND gates to directl implement an SOP epression: One NAND for each Product term One NAND to sum the terms Invert an single inputs Wh do we do this? The NAND integrated circuit design is ver simple. We can use this one simple gate tpe for an epression. The Universit Of Alabama in Huntsville Computer Science SOP to NAND (5) Eample: f(a,b,c,d) = A B + AC + ABD A B A C A B D f(a,b,c,d) 16

17 The Universit Of Alabama in Huntsville Computer Science Cascading 2-input NANDs to implement larger NAND functions Think of a multiple-input NAND as an AND followed b an INVERTER () () We can easil build a multiple-input AND out of 2-input ANDs And we know how to build a 2-input AND from 2-input NANDs Substituting ( () ) = () () = What happened to the Inverters at the output? 17

Boolean Algebra Part 1

Boolean 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 information

BOOLEAN ALGEBRA & LOGIC GATES

BOOLEAN 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 information

CSE140: Midterm 1 Solution and Rubric

CSE140: 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 information

United States Naval Academy Electrical and Computer Engineering Department. EC262 Exam 1

United 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 information

CDA 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 CDA 3200 Digital Systems Instructor: Dr. Janusz Zalewski Developed by: Dr. Dahai Guo Spring 2012 Outline Multi-Level Gate Circuits NAND and NOR Gates Design of Two-Level Circuits Using NAND and NOR Gates

More information

Karnaugh Maps & Combinational Logic Design. ECE 152A Winter 2012

Karnaugh 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 information

1. True or False? A voltage level in the range 0 to 2 volts is interpreted as a binary 1.

1. 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 information

Unit 3 Boolean Algebra (Continued)

Unit 3 Boolean Algebra (Continued) Unit 3 Boolean Algebra (Continued) 1. Exclusive-OR Operation 2. Consensus Theorem Department of Communication Engineering, NCTU 1 3.1 Multiplying Out and Factoring Expressions Department of Communication

More information

Gates, Circuits, and Boolean Algebra

Gates, 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 information

Systems of Linear Equations: Solving by Substitution

Systems of Linear Equations: Solving by Substitution 8.3 Sstems of Linear Equations: Solving b Substitution 8.3 OBJECTIVES 1. Solve sstems using the substitution method 2. Solve applications of sstems of equations In Sections 8.1 and 8.2, we looked at graphing

More information

Elementary Logic Gates

Elementary 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 information

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

Boolean Algebra (cont d) UNIT 3 BOOLEAN ALGEBRA (CONT D) Guidelines for Multiplying Out and Factoring. Objectives. Iris Hui-Ru Jiang Spring 2010 Boolean Algebra (cont d) 2 Contents Multiplying out and factoring expressions Exclusive-OR and Exclusive-NOR operations The consensus theorem Summary of algebraic simplification Proving validity of an

More information

Karnaugh Maps. Circuit-wise, this leads to a minimal two-level implementation

Karnaugh Maps. Circuit-wise, this leads to a minimal two-level 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 information

CSEE 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 CSEE 3827: Fundamentals of Computer Systems Standard Forms and Simplification with Karnaugh Maps Agenda (M&K 2.3-2.5) Standard Forms Product-of-Sums (PoS) Sum-of-Products (SoP) converting between Min-terms

More information

CH3 Boolean Algebra (cont d)

CH3 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. Exclusive-OR and Equivalence operations

More information

2.0 Chapter Overview. 2.1 Boolean Algebra

2.0 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 information

MATH 102 College Algebra

MATH 102 College Algebra FACTORING Factoring polnomials ls is simpl the reverse process of the special product formulas. Thus, the reverse process of special product formulas will be used to factor polnomials. To factor polnomials

More information

Basic Logic Gates Richard E. Haskell

Basic 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 information

Boolean Algebra. Boolean Algebra. Boolean Algebra. Boolean Algebra

Boolean 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 information

FORDHAM UNIVERSITY CISC 3593. Dept. of Computer and Info. Science Spring, 2011. The Binary Adder

FORDHAM UNIVERSITY CISC 3593. Dept. of Computer and Info. Science Spring, 2011. The Binary Adder FORDHAM UNIVERITY CIC 3593 Fordham College Lincoln Center Computer Organization Dept. of Computer and Info. cience pring, 2011 1 Introduction The Binar Adder The binar adder circuit is an important building

More information

Section 7.2 Linear Programming: The Graphical Method

Section 7.2 Linear Programming: The Graphical Method Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function

More information

Zero and Negative Exponents and Scientific Notation. a a n a m n. Now, suppose that we allow m to equal n. We then have. a am m a 0 (1) a m

Zero and Negative Exponents and Scientific Notation. a a n a m n. Now, suppose that we allow m to equal n. We then have. a am m a 0 (1) a m 0. E a m p l e 666SECTION 0. OBJECTIVES. Define the zero eponent. Simplif epressions with negative eponents. Write a number in scientific notation. Solve an application of scientific notation We must have

More information

Simplifying Logic Circuits with Karnaugh Maps

Simplifying 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 information

SECTION 2.2. Distance and Midpoint Formulas; Circles

SECTION 2.2. Distance and Midpoint Formulas; Circles SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation

More information

SECTION 5-1 Exponential Functions

SECTION 5-1 Exponential Functions 354 5 Eponential and Logarithmic Functions Most of the functions we have considered so far have been polnomial and rational functions, with a few others involving roots or powers of polnomial or rational

More information

LESSON EIII.E EXPONENTS AND LOGARITHMS

LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential

More information

5.2 Inverse Functions

5.2 Inverse Functions 78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,

More information

Logic in Computer Science: Logic Gates

Logic 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 information

Introduction to Matrices for Engineers

Introduction to Matrices for Engineers Introduction to Matrices for Engineers C.T.J. Dodson, School of Mathematics, Manchester Universit 1 What is a Matrix? A matrix is a rectangular arra of elements, usuall numbers, e.g. 1 0-8 4 0-1 1 0 11

More information

Lecture 5: Gate Logic Logic Optimization

Lecture 5: Gate Logic Logic Optimization Lecture 5: Gate Logic Logic Optimization MAH, AEN EE271 Lecture 5 1 Overview Reading McCluskey, Logic Design Principles- or any text in boolean algebra Introduction We could design at the level of irsim

More information

Switching Algebra and Logic Gates

Switching 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 information

Understanding Logic Design

Understanding 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 information

Gates & Boolean Algebra. Boolean Operators. Combinational Logic. Introduction

Gates & Boolean Algebra. Boolean Operators. Combinational Logic. Introduction Introduction Gates & Boolean lgebra Boolean algebra: named after mathematician George Boole (85 864). 2-valued algebra. digital circuit can have one of 2 values. Signal between and volt =, between 4 and

More information

THE POWER RULES. Raising an Exponential Expression to a Power

THE POWER RULES. Raising an Exponential Expression to a Power 8 (5-) Chapter 5 Eponents and Polnomials 5. THE POWER RULES In this section Raising an Eponential Epression to a Power Raising a Product to a Power Raising a Quotient to a Power Variable Eponents Summar

More information

exclusive-or and Binary Adder R eouven Elbaz reouven@uwaterloo.ca Office room: DC3576

exclusive-or and Binary Adder R eouven Elbaz reouven@uwaterloo.ca Office room: DC3576 exclusive-or 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 information

Use order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS

Use order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS ORDER OF OPERATIONS In the following order: 1) Work inside the grouping smbols such as parenthesis and brackets. ) Evaluate the powers. 3) Do the multiplication and/or division in order from left to right.

More information

CSE140: Components and Design Techniques for Digital Systems

CSE140: 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 information

Logic Reference Guide

Logic 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 information

Graphing Linear Equations

Graphing Linear Equations 6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are

More information

Digital Logic Design. Basics Combinational Circuits Sequential Circuits. Pu-Jen Cheng

Digital Logic Design. Basics Combinational Circuits Sequential Circuits. Pu-Jen Cheng Digital Logic Design Basics Combinational Circuits Sequential Circuits Pu-Jen Cheng Adapted from the slides prepared by S. Dandamudi for the book, Fundamentals of Computer Organization and Design. Introduction

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,

More information

Digital circuits make up all computers and computer systems. The operation of digital circuits is based on

Digital 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

Higher. Polynomials and Quadratics 64

Higher. Polynomials and Quadratics 64 hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining

More information

ENGI 241 Experiment 5 Basic Logic Gates

ENGI 241 Experiment 5 Basic Logic Gates ENGI 24 Experiment 5 Basic Logic Gates OBJECTIVE This experiment will examine the operation of the AND, NAND, OR, and NOR logic gates and compare the expected outputs to the truth tables for these devices.

More information

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1 Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.

More information

Polynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter

Polynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will

More information

Chapter 1. Computation theory

Chapter 1. Computation theory Chapter 1. Computation theory In this chapter we will describe computation logic for the machines. This topic is a wide interdisciplinary field, so that the students can work in an interdisciplinary context.

More information

CHAPTER 7: FACTORING POLYNOMIALS

CHAPTER 7: FACTORING POLYNOMIALS CHAPTER 7: FACTORING POLYNOMIALS FACTOR (noun) An of two or more quantities which form a product when multiplied together. 1 can be rewritten as 3*, where 3 and are FACTORS of 1. FACTOR (verb) - To factor

More information

C3: Functions. Learning objectives

C3: Functions. Learning objectives CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the

More information

Direct Variation. 1. Write an equation for a direct variation relationship 2. Graph the equation of a direct variation relationship

Direct Variation. 1. Write an equation for a direct variation relationship 2. Graph the equation of a direct variation relationship 6.5 Direct Variation 6.5 OBJECTIVES 1. Write an equation for a direct variation relationship 2. Graph the equation of a direct variation relationship Pedro makes $25 an hour as an electrician. If he works

More information

Karnaugh Maps (K-map) Alternate representation of a truth table

Karnaugh Maps (K-map) Alternate representation of a truth table Karnaugh Maps (K-map) 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 information

Solving Systems of Equations

Solving Systems of Equations Solving Sstems of Equations When we have or more equations and or more unknowns, we use a sstem of equations to find the solution. Definition: A solution of a sstem of equations is an ordered pair that

More information

MAT188H1S Lec0101 Burbulla

MAT188H1S Lec0101 Burbulla Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u

More information

Slope-Intercept Form and Point-Slope Form

Slope-Intercept Form and Point-Slope Form Slope-Intercept Form and Point-Slope Form In this section we will be discussing Slope-Intercept Form and the Point-Slope Form of a line. We will also discuss how to graph using the Slope-Intercept Form.

More information

7.3 Solving Systems by Elimination

7.3 Solving Systems by Elimination 7. Solving Sstems b Elimination In the last section we saw the Substitution Method. It turns out there is another method for solving a sstem of linear equations that is also ver good. First, we will need

More information

Chapter 6 Quadratic Functions

Chapter 6 Quadratic Functions Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. CHAPTER3 QUESTIONS MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. ) If one input of an AND gate is LOW while the other is a clock signal, the output

More information

Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x

Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its

More information

Two-level logic using NAND gates

Two-level logic using NAND gates CSE140: Components and Design Techniques for Digital Systems Two and Multilevel logic implementation Tajana Simunic Rosing 1 Two-level logic using NND gates Replace minterm ND gates with NND gates Place

More information

Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science

Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Colorado College has two all college requirements (QR and SI) which can be satisfied in full, or part, b taking

More information

So, using the new notation, P X,Y (0,1) =.08 This is the value which the joint probability function for X and Y takes when X=0 and Y=1.

So, using the new notation, P X,Y (0,1) =.08 This is the value which the joint probability function for X and Y takes when X=0 and Y=1. Joint probabilit is the probabilit that the RVs & Y take values &. like the PDF of the two events, and. We will denote a joint probabilit function as P,Y (,) = P(= Y=) Marginal probabilit of is the probabilit

More information

Section V.2: Magnitudes, Directions, and Components of Vectors

Section V.2: Magnitudes, Directions, and Components of Vectors Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions

More information

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its

More information

SECTION 7-4 Algebraic Vectors

SECTION 7-4 Algebraic Vectors 7-4 lgebraic Vectors 531 SECTIN 7-4 lgebraic Vectors From Geometric Vectors to lgebraic Vectors Vector ddition and Scalar Multiplication Unit Vectors lgebraic Properties Static Equilibrium Geometric vectors

More information

ELEC 2210 - EXPERIMENT 1 Basic Digital Logic Circuits

ELEC 2210 - EXPERIMENT 1 Basic Digital Logic Circuits Objectives ELEC - EXPERIMENT Basic Digital Logic Circuits The experiments in this laboratory exercise will provide an introduction to digital electronic circuits. You will learn how to use the IDL-00 Bit

More information

Introduction to Digital Logic with Laboratory Exercises

Introduction to Digital Logic with Laboratory Exercises Introduction to Digital Logic with Laboratory Exercises Introduction to Digital Logic with Laboratory Exercises James Feher Copyright 29 James Feher Editor-In-Chief: James Feher Associate Editor: Marisa

More information

Example 1: Model A Model B Total Available. Gizmos. Dodads. System:

Example 1: Model A Model B Total Available. Gizmos. Dodads. System: Lesson : Sstems of Equations and Matrices Outline Objectives: I can solve sstems of three linear equations in three variables. I can solve sstems of linear inequalities I can model and solve real-world

More information

Introduction. The Quine-McCluskey Method Handout 5 January 21, 2016. CSEE E6861y Prof. Steven Nowick

Introduction. The Quine-McCluskey Method Handout 5 January 21, 2016. CSEE E6861y Prof. Steven Nowick CSEE E6861y Prof. Steven Nowick The Quine-McCluskey Method Handout 5 January 21, 2016 Introduction The Quine-McCluskey method is an exact algorithm which finds a minimum-cost sum-of-products implementation

More information

2 Elements of classical computer science

2 Elements of classical computer science 2 Elements of classical computer science 11 Good references for the material of this section are [3], Chap. 3, [5], Secs. 2 and 3, [7], Sec. 6.1, and [11] as a nice readable account of complexit with some

More information

Algebraic Properties and Proofs

Algebraic 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 information

y 1 x dx ln x y a x dx 3. y e x dx e x 15. y sinh x dx cosh x y cos x dx sin x y csc 2 x dx cot x 7. y sec 2 x dx tan x 9. y sec x tan x dx sec x

y 1 x dx ln x y a x dx 3. y e x dx e x 15. y sinh x dx cosh x y cos x dx sin x y csc 2 x dx cot x 7. y sec 2 x dx tan x 9. y sec x tan x dx sec x Strateg for Integration As we have seen, integration is more challenging than differentiation. In finding the derivative of a function it is obvious which differentiation formula we should appl. But it

More information

MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60

MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets

More information

Graphing Quadratic Equations

Graphing Quadratic Equations .4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations

More information

Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.

Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system. _.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial

More information

Affine Transformations

Affine Transformations A P P E N D I X C Affine Transformations CONTENTS C The need for geometric transformations 335 C2 Affine transformations 336 C3 Matri representation of the linear transformations 338 C4 Homogeneous coordinates

More information

Mathematical goals. Starting points. Materials required. Time needed

Mathematical goals. Starting points. Materials required. Time needed Level A7 of challenge: C A7 Interpreting functions, graphs and tables tables Mathematical goals Starting points Materials required Time needed To enable learners to understand: the relationship between

More information

Addition and Subtraction of Vectors

Addition and Subtraction of Vectors ddition and Subtraction of Vectors 1 ppendi ddition and Subtraction of Vectors In this appendi the basic elements of vector algebra are eplored. Vectors are treated as geometric entities represented b

More information

Teaching Digital Circuit Design to Middle-School Students

Teaching Digital Circuit Design to Middle-School Students Paper ID #650 Teaching Digital Circuit Design to Middle-School Students Dr. Christopher R. Carroll, Universit of Minnesota Duluth Dr. Carroll received his undergraduate education at Georgia Tech, and received

More information

Name Date. Break-Even Analysis

Name Date. Break-Even Analysis Name Date Break-Even Analsis In our business planning so far, have ou ever asked the questions: How much do I have to sell to reach m gross profit goal? What price should I charge to cover m costs and

More information

SAMPLE. Polynomial functions

SAMPLE. Polynomial functions Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through

More information

Digital Electronics Detailed Outline

Digital Electronics Detailed Outline Digital Electronics Detailed Outline Unit 1: Fundamentals of Analog and Digital Electronics (32 Total Days) Lesson 1.1: Foundations and the Board Game Counter (9 days) 1. Safety is an important concept

More information

Chapter 7 Memory and Programmable Logic

Chapter 7 Memory and Programmable Logic NCNU_2013_DD_7_1 Chapter 7 Memory and Programmable Logic 71I 7.1 Introduction ti 7.2 Random Access Memory 7.3 Memory Decoding 7.5 Read Only Memory 7.6 Programmable Logic Array 77P 7.7 Programmable Array

More information

DESIGN OF GATE NETWORKS

DESIGN OF GATE NETWORKS DESIGN OF GATE NETWORKS DESIGN OF TWO-LEVEL NETWORKS: and-or and or-and NETWORKS MINIMAL TWO-LEVEL NETWORKS KARNAUGH MAPS MINIMIZATION PROCEDURE AND TOOLS LIMITATIONS OF TWO-LEVEL NETWORKS DESIGN OF TWO-LEVEL

More information

Incremental Reduction of Binary Decision Diagrams

Incremental Reduction of Binary Decision Diagrams Abstract Incremental Reduction of Binar Decision Diagrams R. Jacobi, N. Calazans, C. Trullemans. Université de Louvain Laboratoire de Microélectronique - Place du Levant, 3 B1348 Louvain-la-Neuve Belgium

More information

Core Maths C2. Revision Notes

Core Maths C2. Revision Notes Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...

More information

www.mohandesyar.com SOLUTIONS MANUAL DIGITAL DESIGN FOURTH EDITION M. MORRIS MANO California State University, Los Angeles MICHAEL D.

www.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 information

We start with the basic operations on polynomials, that is adding, subtracting, and multiplying.

We start with the basic operations on polynomials, that is adding, subtracting, and multiplying. R. Polnomials In this section we want to review all that we know about polnomials. We start with the basic operations on polnomials, that is adding, subtracting, and multipling. Recall, to add subtract

More information

4.1 Ordinal versus cardinal utility

4.1 Ordinal versus cardinal utility Microeconomics I. Antonio Zabalza. Universit of Valencia 1 Micro I. Lesson 4. Utilit In the previous lesson we have developed a method to rank consistentl all bundles in the (,) space and we have introduced

More information

Functions and Graphs CHAPTER INTRODUCTION. The function concept is one of the most important ideas in mathematics. The study

Functions and Graphs CHAPTER INTRODUCTION. The function concept is one of the most important ideas in mathematics. The study Functions and Graphs CHAPTER 2 INTRODUCTION The function concept is one of the most important ideas in mathematics. The stud 2-1 Functions 2-2 Elementar Functions: Graphs and Transformations 2-3 Quadratic

More information

Combinational Logic Design

Combinational Logic Design Chapter 4 Combinational Logic Design The foundations for the design of digital logic circuits were established in the preceding chapters. The elements of Boolean algebra (two-element switching algebra

More information

A single register, called the accumulator, stores the. operand before the operation, and stores the result. Add y # add y from memory to the acc

A 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. Accumulator-based 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 information

Vocabulary. Term Page Definition Clarifying Example. biconditional statement. conclusion. conditional statement. conjecture.

Vocabulary. Term Page Definition Clarifying Example. biconditional statement. conclusion. conditional statement. conjecture. CHAPTER Vocabulary The table contains important vocabulary terms from Chapter. As you work through the chapter, fill in the page number, definition, and a clarifying example. biconditional statement conclusion

More information

Combinational Logic Design Process

Combinational Logic Design Process Combinational Logic Design Process Create truth table from specification Generate K-maps & obtain logic equations Draw logic diagram (sharing common gates) Simulate circuit for design verification Debug

More information

CIRCLE THEOREMS. Edexcel GCSE Mathematics (Linear) 1MA0

CIRCLE 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 information

Systems I: Computer Organization and Architecture

Systems I: Computer Organization and Architecture Systems I: Computer Organization and Architecture Lecture 9 - Register Transfer and Microoperations Microoperations Digital systems are modular in nature, with modules containing registers, decoders, arithmetic

More information

Complex Numbers. w = f(z) z. Examples

Complex Numbers. w = f(z) z. Examples omple Numbers Geometrical Transformations in the omple Plane For functions of a real variable such as f( sin, g( 2 +2 etc ou are used to illustrating these geometricall, usuall on a cartesian graph. If

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA. Tuesday, January 24, 2012 9:15 a.m. to 12:15 p.m.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA. Tuesday, January 24, 2012 9:15 a.m. to 12:15 p.m. INTEGRATED ALGEBRA The Universit of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA Tuesda, Januar 4, 01 9:15 a.m. to 1:15 p.m., onl Student Name: School Name: Print our name and

More information

Solutions of Linear Equations in One Variable

Solutions of Linear Equations in One Variable 2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools

More information

CHAPTER 3 Boolean Algebra and Digital Logic

CHAPTER 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 information

SECTION P.5 Factoring Polynomials

SECTION P.5 Factoring Polynomials BLITMCPB.QXP.0599_48-74 /0/0 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises Critical Thinking Eercises 98. The common cold is caused by a rhinovirus. The

More information