Size: px
Start display at page:

Transcription

1 Summary Boolean Addition In Boolean algebra, a variable is a symbol used to represent an action, a condition, or data. A single variable can only have a value of or 0. The complement represents the inverse of a variable and is indicated with an overbar. Thus, the complement of A is A. A literal is a variable or its complement. Addition is equivalent to the OR operation. The sum term is if one or more if the literals are. The sum term is zero only if each literal is 0. Determine the values of A, B, and C that make the sum term of the expression A + B + C = 0? Each literal must = 0; therefore A =, B = 0 and C =.

2 Boolean Multiplication Summary In Boolean algebra, multiplication is equivalent to the AND operation. The product of literals forms a product term. The product term will be only if all of the literals are. What are the values of the A, B and C if the product term of A. B. C =? Each literal must = ; therefore A =, B = 0 and C = 0.

3 Summary Commutative Laws The commutative laws are applied to addition and multiplication. For addition, the commutative law states In terms of the result, the order in which variables are ORed makes no difference. A + B = B + A For multiplication, the commutative law states In terms of the result, the order in which variables are ANDed makes no difference. = BA

4 Summary Associative Laws The associative laws are also applied to addition and multiplication. For addition, the associative law states When ORing more than two variables, the result is the same regardless of the grouping of the variables. A + (B +C) = (A + B) + C For multiplication, the associative law states When ANDing more than two variables, the result is the same regardless of the grouping of the variables. A(BC) = ()C

5 Summary Distributive Law The distributive law is the factoring law. A common variable can be factored from an expression just as in ordinary algebra. That is + AC = A(B+ C) The distributive law can be illustrated with equivalent circuits: B C A B + C A(B+ C) X A B A C AC + AC X

6 Rules of Boolean Algebra Summary. A + 0 = A 2. A + = 3. A. 0 = 0 4. A. = 5. A + A = A 6. A + A = 7. A. A = A 8. A. A = 0 = 9. A = A 0. A + = A. A + = A + B 2. (A + B)(A + C) = A + BC

7 Rules of Boolean Algebra Summary Rules of Boolean algebra can be illustrated with Venn diagrams. The variable A is shown as an area. The rule A + = A can be illustrated easily with a diagram. Add an overlapping area to represent the variable B. The overlap region between A and B represents. A B = A The diagram visually shows that A + = A. Other rules can be illustrated with the diagrams as well.

8 Summary Rules of Boolean Algebra Illustrate the rule diagram. A + = A + Bwith a Venn This time, A is represented by the blue area and B again by the red circle. The intersection represents. Notice that A + = A + B A A BA

9 Rules of Boolean Algebra Summary Rule 2, which states that (A + B)(A + C) = A + BC, can be proven by applying earlier rules as follows: (A + B)(A + C) = AA + AC + + BC = A + AC + + BC = A( + C + B) + BC = A. + BC = A + BC This rule is a little more complicated, but it can also be shown with a Venn diagram, as given on the following slide

10 Summary Three areas represent the variables A, B, and C. The area representing A + B is shown in yellow. The area representing A + C is shown in red. The overlap of red and yellow is shown in orange. The overlapping area between B and C represents BC. ORing with A gives the same area as before. (A + B)(A + C) A B A + B A + C C A B = BC C A + BC

11 DeMorgan s Theorem DeMorgan s st Theorem Summary The complement of a product of variables is equal to the sum of the complemented variables. = A + B Applying DeMorgan s first theorem to gates: A B A B A + B Inputs Output A B A + B NAND Negative-OR

12 DeMorgan s Theorem Summary DeMorgan s 2 nd Theorem The complement of a sum of variables is equal to the product of the complemented variables. A + B = A. B Applying DeMorgan s second theorem to gates: A B NOR A + B A B Negative-AND Inputs Output A B A + B

13 DeMorgan s Theorem Summary Apply DeMorgan s theorem to remove the overbar covering both terms from the expression X = C + D. To apply DeMorgan s theorem to the expression, you can break the overbar covering both terms and change the sign between the terms. This results in = X = C. D. Deleting the double bar gives X = C. D.

14 A B C D Summary Boolean Analysis of Logic Circuits Combinational logic circuits can be analyzed by writing the expression for each gate and combining the expressions according to the rules for Boolean algebra. Apply Boolean algebra to derive the expression for X. Write the expression for each gate: (A + B ) C (A + B ) Applying DeMorgan s theorem and the distribution law: X = C (A B) + D = A B C + D X = C (A + B )+ D

15 Summary Boolean Analysis of Logic Circuits Use Multisim to generate the truth table for the circuit in the previous example. Set up the circuit using the Logic Converter as shown. (Note that the logic converter has no real-world counterpart.) Double-click the Logic Converter top open it. Then click on the conversion bar on the right side to see the truth table for the circuit (see next slide).

16 Summary Boolean Analysis of Logic Circuits The simplified logic expression can be viewed by clicking Simplified expression

17 Summary SOP and POS forms Boolean expressions can be written in the sum-of-products form (SOP) or in the product-of-sums form (POS). These forms can simplify the implementation of combinational logic, particularly with PLDs. In both forms, an overbar cannot extend over more than one variable. An expression is in SOP form when two or more product terms are summed as in the following examples: A B C + A B A B C + C D C D + E An expression is in POS form when two or more sum terms are multiplied as in the following examples: (A + B)(A + C) (A + B + C)(B + D) (A + B)C

18 Summary SOP Standard form In SOP standard form, every variable in the domain must appear in each term. This form is useful for constructing truth tables or for implementing logic in PLDs. You can expand a nonstandard term to standard form by multiplying the term by a term consisting of the sum of the missing variable and its complement. Convert X = A B + A B C to standard form. The first term does not include the variable C. Therefore, multiply it by the (C + C), which = : X = A B (C + C) + A B C = A B C + A B C + A B C

19 SOP Standard form Summary The Logic Converter in Multisim can convert a circuit into standard SOP form. Use Multisim to view the logic for the circuit in standard SOP form. Click the truth table to logic button on the Logic Converter. See next slide

20 Summary SOP Standard form SOP Standard form

21 Summary POS Standard form In POS standard form, every variable in the domain must appear in each sum term of the expression. You can expand a nonstandard POS expression to standard form by adding the product of the missing variable and its complement and applying rule 2, which states that (A + B)(A + C) = A + BC. Convert X = (A + B)(A + B + C) to standard form. The first sum term does not include the variable C. Therefore, add C C and expand the result by rule 2. X = (A + B + C C)(A + B + C) = (A +B + C )(A + B + C)(A + B + C)

22 Karnaugh maps Summary The Karnaugh map (K-map) is a tool for simplifying combinational logic with 3 or 4 variables. For 3 variables, 8 cells are required (2 3 ). The map shown is for three variables labeled A, B, and C. Each cell represents one possible product term. Each cell differs from an adjacent cell by only one variable. C C C C C C C C

23 Summary Gray code Karnaugh maps Cells are usually labeled using 0 s and s to represent the variable and its complement. C The numbers are entered in gray code, to force adjacent cells to be different by only one variable. Ones are read as the true variable and zeros are read as the complemented variable.

24 Summary Karnaugh maps Alternatively, cells can be labeled with the variable letters. This makes it simple to read, but it takes more time preparing the map. Read the terms for the yellow cells. C C C C C C C C The cells are C and C. C C C C

25 Summary Karnaugh maps K-maps can simplify combinational logic by grouping cells and eliminating variables that change. Group the s on the map and read the minimum logic. B changes across this boundary C C changes across this boundary. Group the s into two overlapping groups as indicated. 2. Read each group by eliminating any variable that changes across a boundary. 3. The vertical group is read AC. 4. The horizontal group is read. X = AC +

26 Summary Karnaugh maps A 4-variable map has an adjacent cell on each of its four boundaries as shown. CD CD CD CD Each cell is different only by one variable from an adjacent cell. Grouping follows the rules given in the text. The following slide shows an example of reading a four variable map using binary numbers for the variables

27 Summary Karnaugh maps Group the s on the map and read the minimum logic. B changes B changes CD C changes across outer boundary C changes X. Group the s into two separate groups as indicated. 2. Read each group by eliminating any variable that changes across a boundary. 3. The upper (yellow) group is read as AD. 4. The lower (green) group is read as AD. X = AD +AD

28 Summary Hardware Description Languages (HDLs) A Hardware Description Language (HDL) is a tool for implementing a logic design in a PLD. One important language is called VHDL. In VHDL, there are three approaches to describing logic:. Structural Description is like a schematic (components and block diagrams). 2. Dataflow 3. Behavioral Description is equations, such as Boolean operations, and registers. Description is specifications over time (state machines, etc.).

29 Summary Hardware Description Languages (HDLs) The data flow method for VHDL uses Boolean-type statements. There are two-parts to a basic data flow program: the entity and the architecture. The entity portion describes the I/O. The architecture portion describes the logic. The following example is a VHDL program showing the two parts. The program is used to detect an invalid BCD code. entity BCDInv is port (B,C,D: in bit; X: out bit); end entity BCDInv architecture Invalid of BCDInv begin X <= (B or C) and D; end architecture Invalid;

30 Summary Hardware Description Languages (HDLs) Another standard HDL is Verilog. In Verilog, the I/O and the logic is described in one unit called a module. Verilog uses specific symbols to stand for the Boolean logical operators. The following is the same program as in the previous slide, written for Verilog: module BCDInv (X, B, C, D); input B, C, D; output X; assign X = (B C)&D; endmodule

31 Selected Key Terms Variable Complement Sum term Product term A symbol used to represent a logical quantity that can have a value of or 0, usually designated by an italic letter. The inverse or opposite of a number. In Boolean algebra, the inverse function, expressed with a bar over the variable. The Boolean sum of two or more literals equivalent to an OR operation. The Boolean product of two or more literals equivalent to an AND operation.

32 Selected Key Terms Sum-ofproducts (SOP) Product of sums (POS) Karnaugh map VHDL A form of Boolean expression that is basically the ORing of ANDed terms. A form of Boolean expression that is basically the ANDing of ORed terms. An arrangement of cells representing combinations of literals in a Boolean expression and used for systematic simplification of the expression. A standard hardware description language. IEEE Std

33 . The associative law for addition is normally written as a. A + B = B + A b. (A + B) + C = A + (B + C) c. = BA d. A + = A 2008 Pearson Education

34 2. The Boolean equation + AC = A(B+ C) illustrates a. the distribution law b. the commutative law c. the associative law d. DeMorgan s theorem 2008 Pearson Education

35 3. The Boolean expression A. is equal to a. A b. B c. 0 d Pearson Education

36 4. The Boolean expression A + is equal to a. A b. B c. 0 d Pearson Education

37 5. The Boolean equation + AC = A(B+ C) illustrates a. the distribution law b. the commutative law c. the associative law d. DeMorgan s theorem 2008 Pearson Education

38 6. A Boolean expression that is in standard SOP form is a. the minimum logic expression b. contains only one product term c. has every variable in the domain in every term d. none of the above 2008 Pearson Education

39 7. Adjacent cells on a Karnaugh map differ from each other by a. one variable b. two variables c. three variables d. answer depends on the size of the map 2008 Pearson Education

40 8. The minimum expression that can be read from the Karnaugh map shown is a. X = A b. X = A c. X = B d. X = B C C 2008 Pearson Education

41 9. The minimum expression that can be read from the Karnaugh map shown is a. X = A b. X = A c. X = B d. X = B C C 2008 Pearson Education

42 0. In VHDL code, the two main parts are called the a. I/O and the module b. entity and the architecture c. port and the module d. port and the architecture 2008 Pearson Education

### Digital Logic Design (CEN-120) (3+1)

Digital Logic Design (CEN-120) (3+1) ASSISTANT PROFESSOR Engr. Syed Rizwan Ali, MS(CAAD)UK, PDG(CS)UK, PGD(PM)IR, BS(CE)PK HEC Certified Master Trainer (MT-FPDP) PEC Certified Professional Engineer (COM/2531)

### Digital Fundamentals

Digital Fundamentals Tenth Edition Floyd Chapter 3 Modified by Yuttapong Jiraraksopakun Floyd, Digital Fundamentals, th 28 Pearson Education ENE, KMUTT ed 29 The Inverter Summary The inverter performs

### LSN 4 Boolean Algebra & Logic Simplification. ECT 224 Digital Computer Fundamentals. Department of Engineering Technology

LSN 4 Boolean Algebra & Logic Simplification Department of Engineering Technology LSN 4 Key Terms Variable: a symbol used to represent a logic quantity Compliment: the inverse of a variable Literal: a

### Assignment (3-6) Boolean Algebra and Logic Simplification - General Questions

Assignment (3-6) Boolean Algebra and Logic Simplification - General Questions 1. Convert the following SOP expression to an equivalent POS expression. 2. Determine the values of A, B, C, and D that make

### SYNERGY INSTITUTE OF ENGINEERING & TECHNOLOGY,DHENKANAL LECTURE NOTES ON DIGITAL ELECTRONICS CIRCUIT(SUBJECT CODE:PCEC4202)

Lecture No:5 Boolean Expressions and Definitions Boolean Algebra Boolean Algebra is used to analyze and simplify the digital (logic) circuits. It uses only the binary numbers i.e. 0 and 1. It is also called

### DKT 122/3 DIGITAL SYSTEM 1

Company LOGO DKT 122/3 DIGITAL SYSTEM 1 BOOLEAN ALGEBRA (PART 2) Boolean Algebra Contents Boolean Operations & Expression Laws & Rules of Boolean algebra DeMorgan s Theorems Boolean analysis of logic circuits

### Boolean Analysis of Logic Circuits

Course: B.Sc. Applied Physical Science (Computer Science) Year & Sem.: IInd Year, Sem - IIIrd Subject: Computer Science Paper No.: IX Paper Title: Computer System Architecture Lecture No.: 7 Lecture Title:

### Bawar Abid Abdalla. Assistant Lecturer Software Engineering Department Koya University

Logic Design First Stage Lecture No.5 Boolean Algebra Bawar Abid Abdalla Assistant Lecturer Software Engineering Department Koya University Boolean Operations Laws of Boolean Algebra Rules of Boolean Algebra

### IT 201 Digital System Design Module II Notes

IT 201 Digital System Design Module II Notes 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.

### Experiment 4 Boolean Functions Implementation

Experiment 4 Boolean Functions Implementation Introduction: Generally you will find that the basic logic functions AND, OR, NAND, NOR, and NOT are not sufficient to implement complex digital logic functions.

### EEE130 Digital Electronics I Lecture #4_1

EEE130 Digital Electronics I Lecture #4_1 - Boolean Algebra and Logic Simplification - By Dr. Shahrel A. Suandi 4-6 Standard Forms of Boolean Expressions There are two standard forms: Sum-of-products form

### Bawar Abid Abdalla. Assistant Lecturer Software Engineering Department Koya University

Logic Design First Stage Lecture No.6 Boolean Algebra Bawar Abid Abdalla Assistant Lecturer Software Engineering Department Koya University Outlines Boolean Operations Laws of Boolean Algebra Rules of

### Ch. 5 : Boolean Algebra &

Ch. 5 : Boolean Algebra & Reduction elektronik@fisika.ui.ac.id Objectives Should able to: Write Boolean equations for combinational logic applications. Utilize Boolean algebra laws and rules for simplifying

### To write Boolean functions in their standard Min and Max terms format. To simplify Boolean expressions using Karnaugh Map.

3.1 Objectives To write Boolean functions in their standard Min and Max terms format. To simplify Boolean expressions using. 3.2 Sum of Products & Product of Sums Any Boolean expression can be simplified

### Chapter 2. Boolean Expressions:

Chapter 2 Boolean Expressions: A Boolean expression or a function is an expression which consists of binary variables joined by the Boolean connectives AND and OR along with NOT operation. Any Boolean

### Combinational Logic Circuits

Chapter 3 Combinational Logic Circuits 12 Hours 24 Marks 3.1 Standard representation for logical functions Boolean expressions / logic expressions / logical functions are expressed in terms of logical

### ELCT201: DIGITAL LOGIC DESIGN

ELCT201: DIGITAL LOGIC DESIGN Dr. Eng. Haitham Omran, haitham.omran@guc.edu.eg Dr. Eng. Wassim Alexan, wassim.joseph@guc.edu.eg Lecture 3 Following the slides of Dr. Ahmed H. Madian ذو الحجة 1438 ه Winter

### CHAPTER-2 STRUCTURE OF BOOLEAN FUNCTION USING GATES, K-Map and Quine-McCluskey

CHAPTER-2 STRUCTURE OF BOOLEAN FUNCTION USING GATES, K-Map and Quine-McCluskey 2. Introduction Logic gates are connected together to produce a specified output for certain specified combinations of input

### ELCT201: DIGITAL LOGIC DESIGN

ELCT201: DIGITAL LOGIC DESIGN Dr. Eng. Haitham Omran, haitham.omran@guc.edu.eg Dr. Eng. Wassim Alexan, wassim.joseph@guc.edu.eg Lecture 3 Following the slides of Dr. Ahmed H. Madian محرم 1439 ه Winter

### 2.6 BOOLEAN FUNCTIONS

2.6 BOOLEAN FUNCTIONS Binary variables have two values, either 0 or 1. A Boolean function is an expression formed with binary variables, the two binary operators AND and OR, one unary operator NOT, parentheses

### Module -7. Karnaugh Maps

1 Module -7 Karnaugh Maps 1. Introduction 2. Canonical and Standard forms 2.1 Minterms 2.2 Maxterms 2.3 Canonical Sum of Product or Sum-of-Minterms (SOM) 2.4 Canonical product of sum or Product-of-Maxterms(POM)

Get Free notes at Module-I One s Complement: Complement all the bits.i.e. makes all 1s as 0s and all 0s as 1s Two s Complement: One s complement+1 SIGNED BINARY NUMBERS Positive integers (including zero)

### Chapter 2 Combinational

Computer Engineering 1 (ECE290) Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization HOANG Trang 2008 Pearson Education, Inc. Overview Part 1 Gate Circuits and Boolean Equations Binary Logic

### 4 KARNAUGH MAP MINIMIZATION

4 KARNAUGH MAP MINIMIZATION A Karnaugh map provides a systematic method for simplifying Boolean expressions and, if properly used, will produce the simplest SOP or POS expression possible, known as the

### Combinational Circuits Digital Logic (Materials taken primarily from:

Combinational Circuits Digital Logic (Materials taken primarily from: http://www.facstaff.bucknell.edu/mastascu/elessonshtml/eeindex.html http://www.cs.princeton.edu/~cos126 ) Digital Systems What is a

### BOOLEAN ALGEBRA. Logic circuit: 1. From logic circuit to Boolean expression. Derive the Boolean expression for the following circuits.

COURSE / CODE DIGITAL SYSTEMS FUNDAMENTAL (ECE 421) DIGITAL ELECTRONICS FUNDAMENTAL (ECE 422) BOOLEAN ALGEBRA Boolean Logic Boolean logic is a complete system for logical operations. It is used in countless

### Chapter 2 Combinational Logic Circuits

Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization Charles Kime & Thomas Kaminski 2008 Pearson Education, Inc. (Hyperlinks are active in View Show

### Points Addressed in this Lecture. Standard form of Boolean Expressions. Lecture 4: Logic Simplication & Karnaugh Map

Points Addressed in this Lecture Lecture 4: Logic Simplication & Karnaugh Map Professor Peter Cheung Department of EEE, Imperial College London Standard form of Boolean Expressions Sum-of-Products (SOP),

### Boolean relationships on Venn Diagrams

Boolean relationships on Venn Diagrams The fourth example has A partially overlapping B. Though, we will first look at the whole of all hatched area below, then later only the overlapping region. Let's

### Experiment 3: Logic Simplification

Module: Logic Design Name:... University no:.. Group no:. Lab Partner Name: Mr. Mohamed El-Saied Experiment : Logic Simplification Objective: How to implement and verify the operation of the logical functions

### Unit-IV Boolean Algebra

Unit-IV Boolean Algebra Boolean Algebra Chapter: 08 Truth table: Truth table is a table, which represents all the possible values of logical variables/statements along with all the possible results of

### Chapter 2 Combinational Logic Circuits

Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization Overview Part Gate Circuits and Boolean Equations Binary Logic and Gates Boolean Algebra Standard

### Gate Level Minimization Map Method

Gate Level Minimization Map Method Complexity of hardware implementation is directly related to the complexity of the algebraic expression Truth table representation of a function is unique Algebraically

### Chapter 3. Gate-Level Minimization. Outlines

Chapter 3 Gate-Level Minimization Introduction The Map Method Four-Variable Map Five-Variable Map Outlines Product of Sums Simplification Don t-care Conditions NAND and NOR Implementation Other Two-Level

### X Y Z F=X+Y+Z

This circuit is used to obtain the compliment of a value. If X = 0, then X = 1. The truth table for NOT gate is : X X 0 1 1 0 2. OR gate : The OR gate has two or more input signals but only one output

### A graphical method of simplifying logic

4-5 Karnaugh Map Method A graphical method of simplifying logic equations or truth tables. Also called a K map. Theoretically can be used for any number of input variables, but practically limited to 5

### Slide Set 5. for ENEL 353 Fall Steve Norman, PhD, PEng. Electrical & Computer Engineering Schulich School of Engineering University of Calgary

Slide Set 5 for ENEL 353 Fall 207 Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary Fall Term, 207 SN s ENEL 353 Fall 207 Slide Set 5 slide

### CS470: Computer Architecture. AMD Quad Core

CS470: Computer Architecture Yashwant K. Malaiya, Professor malaiya@cs.colostate.edu AMD Quad Core 1 Architecture Layers Building blocks Gates, flip-flops Functional bocks: Combinational, Sequential Instruction

### Digital Logic Design (3)

Digital Logic Design (3) ENGG1015 1 st Semester, 2010 Dr. Kenneth Wong Dr. Hayden So Department of Electrical and Electronic Engineering Last lecture ll logic functions can be represented as (1) truth

### Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world

Pearson Education Limited Edinburgh Gate Harlow Essex M2 2JE England and Associated ompanies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk Pearson Education Limited 24 All

### MODULE 5 - COMBINATIONAL LOGIC

Introduction to Digital Electronics Module 5: Combinational Logic 1 MODULE 5 - COMBINATIONAL LOGIC OVERVIEW: For any given combination of input binary bits or variables, the logic will have a specific

### 1. Mark the correct statement(s)

1. Mark the correct statement(s) 1.1 A theorem in Boolean algebra: a) Can easily be proved by e.g. logic induction b) Is a logical statement that is assumed to be true, c) Can be contradicted by another

### Combinational Devices and Boolean Algebra

Combinational Devices and Boolean Algebra Silvina Hanono Wachman M.I.T. L02-1 6004.mit.edu Home: Announcements, course staff Course information: Lecture and recitation times and locations Course materials

### Menu. Algebraic Simplification - Boolean Algebra EEL3701 EEL3701. MSOP, MPOS, Simplification

Menu Minterms & Maxterms SOP & POS MSOP & MPOS Simplification using the theorems/laws/axioms Look into my... 1 Definitions (Review) Algebraic Simplification - Boolean Algebra Minterms (written as m i ):

### Injntu.com Injntu.com Injntu.com R16

1. a) What are the three methods of obtaining the 2 s complement of a given binary (3M) number? b) What do you mean by K-map? Name it advantages and disadvantages. (3M) c) Distinguish between a half-adder

### Lecture 7 Logic Simplification

Lecture 7 Logic Simplification Simplification Using oolean lgebra simplified oolean expression uses the fewest gates possible to implement a given expression. +(+)+(+) Simplification Using oolean lgebra

### Binary logic. Dr.Abu-Arqoub

Binary logic Binary logic deals with variables like (a, b, c,, x, y) that take on two discrete values (, ) and with operations that assume logic meaning ( AND, OR, NOT) Truth table is a table of all possible

### 9/10/2016. The Dual Form Swaps 0/1 and AND/OR. ECE 120: Introduction to Computing. Every Boolean Expression Has a Dual Form

University of Illinois at Urbana-Champaign Dept. of Electrical and Computer Engineering ECE 120: Introduction to Computing Boolean Properties and Optimization The Dual Form Swaps 0/1 and AND/OR Boolean

### Combinational Logic & Circuits

Week-I Combinational Logic & Circuits Spring' 232 - Logic Design Page Overview Binary logic operations and gates Switching algebra Algebraic Minimization Standard forms Karnaugh Map Minimization Other

### Circuit analysis summary

Boolean Algebra Circuit analysis summary After finding the circuit inputs and outputs, you can come up with either an expression or a truth table to describe what the circuit does. You can easily convert

### Specifying logic functions

CSE4: Components and Design Techniques for Digital Systems Specifying logic functions Instructor: Mohsen Imani Slides from: Prof.Tajana Simunic and Dr.Pietro Mercati We have seen various concepts: Last

### Computer Science. Unit-4: Introduction to Boolean Algebra

Unit-4: Introduction to Boolean Algebra Learning Objective At the end of the chapter students will: Learn Fundamental concepts and basic laws of Boolean algebra. Learn about Boolean expression and will

### Boolean Algebra and Logic Gates

Boolean Algebra and Logic Gates Binary logic is used in all of today's digital computers and devices Cost of the circuits is an important factor Finding simpler and cheaper but equivalent circuits can

### ENGIN 112 Intro to Electrical and Computer Engineering

ENGIN 2 Intro to Electrical and Computer Engineering Lecture 5 Boolean Algebra Overview Logic functions with s and s Building digital circuitry Truth tables Logic symbols and waveforms Boolean algebra

### Combinational Logic Circuits Part III -Theoretical Foundations

Combinational Logic Circuits Part III -Theoretical Foundations Overview Simplifying Boolean Functions Algebraic Manipulation Karnaugh Map Manipulation (simplifying functions of 2, 3, 4 variables) Systematic

### Chapter 2. Boolean Algebra and Logic Gates

Chapter 2. Boolean Algebra and Logic Gates Tong In Oh 1 Basic Definitions 2 3 2.3 Axiomatic Definition of Boolean Algebra Boolean algebra: Algebraic structure defined by a set of elements, B, together

### Date Performed: Marks Obtained: /10. Group Members (ID):. Experiment # 04. Boolean Expression Simplification and Implementation

Name: Instructor: Engr. Date Performed: Marks Obtained: /10 Group Members (ID):. Checked By: Date: Experiment # 04 Boolean Expression Simplification and Implementation OBJECTIVES: To understand the utilization

### CS8803: Advanced Digital Design for Embedded Hardware

CS883: Advanced Digital Design for Embedded Hardware Lecture 2: Boolean Algebra, Gate Network, and Combinational Blocks Instructor: Sung Kyu Lim (limsk@ece.gatech.edu) Website: http://users.ece.gatech.edu/limsk/course/cs883

### Literal Cost F = BD + A B C + A C D F = BD + A B C + A BD + AB C F = (A + B)(A + D)(B + C + D )( B + C + D) L = 10

Circuit Optimization Goal: To obtain the simplest implementation for a given function Optimization is a more formal approach to simplification that is performed using a specific procedure or algorithm

### Simplification of Boolean Functions

COM111 Introduction to Computer Engineering (Fall 2006-2007) NOTES 5 -- page 1 of 5 Introduction Simplification of Boolean Functions You already know one method for simplifying Boolean expressions: Boolean

### Introduction to Computer Architecture

Boolean Operators The Boolean operators AND and OR are binary infix operators (that is, they take two arguments, and the operator appears between them.) A AND B D OR E We will form Boolean Functions of

### Combinational Logic Circuits

Chapter 2 Combinational Logic Circuits J.J. Shann (Slightly trimmed by C.P. Chung) Chapter Overview 2-1 Binary Logic and Gates 2-2 Boolean Algebra 2-3 Standard Forms 2-4 Two-Level Circuit Optimization

### Lecture (05) Boolean Algebra and Logic Gates

Lecture (05) Boolean Algebra and Logic Gates By: Dr. Ahmed ElShafee ١ Minterms and Maxterms consider two binary variables x and y combined with an AND operation. Since eachv ariable may appear in either

### Logic and Computer Design Fundamentals. Chapter 2 Combinational Logic Circuits. Part 3 Additional Gates and Circuits

Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 3 Additional Gates and Circuits Charles Kime & Thomas Kaminski 28 Pearson Education, Inc. (Hyperlinks are active in View

### 2. BOOLEAN ALGEBRA 2.1 INTRODUCTION

2. BOOLEAN ALGEBRA 2.1 INTRODUCTION In the previous chapter, we introduced binary numbers and binary arithmetic. As you saw in binary arithmetic and in the handling of floating-point numbers, there is

### Austin Herring Recitation 002 ECE 200 Project December 4, 2013

1. Fastest Circuit a. How Design Was Obtained The first step of creating the design was to derive the expressions for S and C out from the given truth tables. This was done using Karnaugh maps. The Karnaugh

### Gate-Level Minimization

Gate-Level Minimization ( 范倫達 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C. Fall, 2017 ldvan@cs.nctu.edu.tw http://www.cs.nctu.edu.tw/~ldvan/ Outlines The Map Method

### Gate Level Minimization

Gate Level Minimization By Dr. M. Hebaishy Digital Logic Design Ch- Simplifying Boolean Equations Example : Y = AB + AB Example 2: = B (A + A) T8 = B () T5 = B T Y = A(AB + ABC) = A (AB ( + C ) ) T8 =

### UNIT-4 BOOLEAN LOGIC. NOT Operator Operates on single variable. It gives the complement value of variable.

UNIT-4 BOOLEAN LOGIC Boolean algebra is an algebra that deals with Boolean values((true and FALSE). Everyday we have to make logic decisions: Should I carry the book or not?, Should I watch TV or not?

### R.M.D. ENGINEERING COLLEGE R.S.M. Nagar, Kavaraipettai

L T P C R.M.D. ENGINEERING COLLEGE R.S.M. Nagar, Kavaraipettai- 601206 DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING EC8392 UNIT - I 3 0 0 3 OBJECTIVES: To present the Digital fundamentals, Boolean

### SEE1223: Digital Electronics

SEE223: Digital Electronics 3 Combinational Logic Design Zulkifil Md Yusof Dept. of Microelectronics and Computer Engineering The aculty of Electrical Engineering Universiti Teknologi Malaysia Karnaugh

### Digital Logic Lecture 7 Gate Level Minimization

Digital Logic Lecture 7 Gate Level Minimization By Ghada Al-Mashaqbeh The Hashemite University Computer Engineering Department Outline Introduction. K-map principles. Simplification using K-maps. Don t-care

### Boolean Logic CS.352.F12

Boolean Logic CS.352.F12 Boolean Algebra Boolean Algebra Mathematical system used to manipulate logic equations. Boolean: deals with binary values (True/False, yes/no, on/off, 1/0) Algebra: set of operations

### Logic Design: Part 2

Orange Coast College Business Division Computer Science Department CS 6- Computer Architecture Logic Design: Part 2 Where are we? Number systems Decimal Binary (and related Octal and Hexadecimal) Binary

28 The McGraw-Hill Companies, Inc. All rights reserved. 28 The McGraw-Hill Companies, Inc. All rights reserved. All or Nothing Gate Boolean Expression: A B = Y Truth Table (ee next slide) or AB = Y 28

### DIGITAL SYSTEM DESIGN

DIGITAL SYSTEM DESIGN UNIT I: Introduction to Number Systems and Boolean Algebra Digital and Analog Basic Concepts, Some history of Digital Systems-Introduction to number systems, Binary numbers, Number

### Announcements. Chapter 2 - Part 1 1

Announcements If you haven t shown the grader your proof of prerequisite, please do so by 11:59 pm on 09/05/2018 (Wednesday). I will drop students that do not show us the prerequisite proof after this

### DIGITAL CIRCUIT LOGIC UNIT 5: KARNAUGH MAPS (K-MAPS)

DIGITAL CIRCUIT LOGIC UNIT 5: KARNAUGH MAPS (K-MAPS) 1 Learning Objectives 1. Given a function (completely or incompletely specified) of three to five variables, plot it on a Karnaugh map. The function

### DeMorgan's Theorem. George Self. 1 Introduction

OpenStax-CNX module: m46633 1 DeMorgan's Theorem George Self This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 3.0 Abstract Boolean Algebra is used to mathematically

### Code No: R Set No. 1

Code No: R059210504 Set No. 1 II B.Tech I Semester Supplementary Examinations, February 2007 DIGITAL LOGIC DESIGN ( Common to Computer Science & Engineering, Information Technology and Computer Science

### CSCI 220: Computer Architecture I Instructor: Pranava K. Jha. Simplification of Boolean Functions using a Karnaugh Map

CSCI 22: Computer Architecture I Instructor: Pranava K. Jha Simplification of Boolean Functions using a Karnaugh Map Q.. Plot the following Boolean function on a Karnaugh map: f(a, b, c, d) = m(, 2, 4,

### Chapter 3 Simplification of Boolean functions

3.1 Introduction Chapter 3 Simplification of Boolean functions In this chapter, we are going to discuss several methods for simplifying the Boolean function. What is the need for simplifying the Boolean

### PART 1. Simplification Using Boolean Algebra

Name EET 1131 Lab #5 Logic Simplification Techniques OBJECTIVES: Upon completing this lab, you ll be able to: 1) Obtain the experimental truth table of a logic circuit. 2) Use Boolean algebra to simplify

### Chapter 3. Boolean Algebra and Digital Logic

Chapter 3 Boolean Algebra and Digital Logic Chapter 3 Objectives Understand the relationship between Boolean logic and digital computer circuits. Learn how to design simple logic circuits. Understand how

### Variable, Complement, and Literal are terms used in Boolean Algebra.

We have met gate logic and combination of gates. Another way of representing gate logic is through Boolean algebra, a way of algebraically representing logic gates. You should have already covered the

### Designing Computer Systems Boolean Algebra

Designing Computer Systems Boolean Algebra 08:34:45 PM 4 June 2013 BA-1 Scott & Linda Wills Designing Computer Systems Boolean Algebra Programmable computers can exhibit amazing complexity and generality.

### SWITCHING THEORY AND LOGIC CIRCUITS

SWITCHING THEORY AND LOGIC CIRCUITS COURSE OBJECTIVES. To understand the concepts and techniques associated with the number systems and codes 2. To understand the simplification methods (Boolean algebra

### Chapter 2 Boolean algebra and Logic Gates

Chapter 2 Boolean algebra and Logic Gates 2. Introduction In working with logic relations in digital form, we need a set of rules for symbolic manipulation which will enable us to simplify complex expressions

### ENGIN 112. Intro to Electrical and Computer Engineering

ENIN 2 Intro to Electrical and Computer Engineering Lecture 6 More Boolean Algebra ENIN2 L6: More Boolean Algebra September 5, 23 A B Overview Epressing Boolean functions Relationships between algebraic

### Simplification of Boolean Functions

Simplification of Boolean Functions Contents: Why simplification? The Map Method Two, Three, Four and Five variable Maps. Simplification of two, three, four and five variable Boolean function by Map method.

### Lecture 3: Binary Subtraction, Switching Algebra, Gates, and Algebraic Expressions

EE210: Switching Systems Lecture 3: Binary Subtraction, Switching Algebra, Gates, and Algebraic Expressions Prof. YingLi Tian Feb. 5/7, 2019 Department of Electrical Engineering The City College of New

### Combinational Circuits

Combinational Circuits Combinational circuit consists of an interconnection of logic gates They react to their inputs and produce their outputs by transforming binary information n input binary variables

### Logic Gates and Boolean Algebra ENT263

Logic Gates and Boolean Algebra ENT263 Logic Gates and Boolean Algebra Now that we understand the concept of binary numbers, we will study ways of describing how systems using binary logic levels make

### Lecture 5. Chapter 2: Sections 4-7

Lecture 5 Chapter 2: Sections 4-7 Outline Boolean Functions What are Canonical Forms? Minterms and Maxterms Index Representation of Minterms and Maxterms Sum-of-Minterm (SOM) Representations Product-of-Maxterm

### Digital Techniques. Lecture 1. 1 st Class

Digital Techniques Lecture 1 1 st Class Digital Techniques Digital Computer and Digital System: Digital computer is a part of digital system, it based on binary system. A block diagram of digital computer

### ENGINEERS ACADEMY. 7. Given Boolean theorem. (a) A B A C B C A B A C. (b) AB AC BC AB BC. (c) AB AC BC A B A C B C.

Digital Electronics Boolean Function QUESTION BANK. The Boolean equation Y = C + C + C can be simplified to (a) (c) A (B + C) (b) AC (d) C. The Boolean equation Y = (A + B) (A + B) can be simplified to

### Gate-Level Minimization. BME208 Logic Circuits Yalçın İŞLER

Gate-Level Minimization BME28 Logic Circuits Yalçın İŞLER islerya@yahoo.com http://me.islerya.com Complexity of Digital Circuits Directly related to the complexity of the algebraic expression we use to

### Gate-Level Minimization

MEC520 디지털공학 Gate-Level Minimization Jee-Hwan Ryu School of Mechanical Engineering Gate-Level Minimization-The Map Method Truth table is unique Many different algebraic expression Boolean expressions may

### QUESTION BANK FOR TEST

CSCI 2121 Computer Organization and Assembly Language PRACTICE QUESTION BANK FOR TEST 1 Note: This represents a sample set. Please study all the topics from the lecture notes. Question 1. Multiple Choice

### EECS150 Homework 2 Solutions Fall ) CLD2 problem 2.2. Page 1 of 15

1.) CLD2 problem 2.2 We are allowed to use AND gates, OR gates, and inverters. Note that all of the Boolean expression are already conveniently expressed in terms of AND's, OR's, and inversions. Thus,