Digital Fundamentals

Size: px
Start display at page:

Download "Digital Fundamentals"

Transcription

1 Digital Fundamentals Tenth Edition Floyd Chapter 3 Modified by Yuttapong Jiraraksopakun Floyd, Digital Fundamentals, th 28 Pearson Education ENE, KMUTT ed 29

2 The Inverter Summary The inverter performs the oolean NOT operation. When the input is LOW, the output is HIGH; when the input is HIGH, the output is LOW. Input Output LOW () HIGH () HIGH () LOW() The NOT operation (complement) is shown with an overbar. Thus, the oolean expression for an inverter is =.

3 The Inverter Summary Example waveforms: group of inverters can be used to form the s complement of a binary number: inary number s complement

4 The ND Gate Summary The ND gate produces a HIGH output when all inputs are HIGH; otherwise, the output is LOW. For a 2-input gate, the truth table is Inputs Output The ND operation is usually shown with a dot between the variables but it may be implied (no dot). Thus, the ND operation is written as =. or =. &

5 Summary The ND Gate & Example waveforms: The ND operation is used in computer programming as a selective mask. If you want to retain certain bits of a binary number but reset the other bits to, you could set a mask with s in the position of the retained bits. If the binary number is NDed with the mask, what is the result?

6 The OR Gate Summary The OR gate produces a HIGH output if any input is HIGH; if all inputs are LOW, the output is LOW. For a 2-input gate, the truth table is Inputs Output The OR operation is shown with a plus sign (+) between the variables. Thus, the OR operation is written as = +.

7 Summary The OR Gate Example waveforms: The OR operation can be used in computer programming to set certain bits of a binary number to. SCII letters have a in the bit 5 position for lower case letters and a in this position for capitals. (it positions are numbered from right to left starting with.) What will be the result if you OR an SCII letter with the 8-bit mask? The resulting letter will be lower case.

8 The NND Gate Summary The NND gate produces a LOW output when all inputs are HIGH; otherwise, the output is HIGH. For a 2-input gate, the truth table is Inputs Output The NND operation is shown with a dot between the variables and an overbar covering them. Thus, the NND operation is written as =. (lternatively, =.) &

9 Summary The NND Gate & Example waveforms: The NND gate is particularly useful because it is a universal gate all other basic gates can be constructed from NND gates. How would you connect a 2-input NND gate to form a basic inverter?

10 Negative-OR Equivalent Summary

11 The NOR Gate Summary The NOR gate produces a LOW output if any input is HIGH; if all inputs are HIGH, the output is LOW. For a 2-input gate, the truth table is Inputs Output The NOR operation is shown with a plus sign (+) between the variables and an overbar covering them. Thus, the NOR operation is written as = +.

12 Summary The NOR Gate Example waveforms: The NOR operation will produce a LOW if any input is HIGH. When is the LED is ON for the circuit shown? +5. V 33 Ω The LED will be on when any of the four inputs are HIGH. C D

13 Negative-ND Equivalent Summary

14 The OR Gate Summary The OR gate produces a HIGH output only when both inputs are at opposite logic levels. The truth table is = Inputs Output The OR operation is written as = +. lternatively, it can be written with a circled plus sign between the variables as = +.

15 Summary The OR Gate = Example waveforms: Notice that the OR gate will produce a HIGH only when exactly one input is HIGH. If the and waveforms are both inverted for the above waveforms, how is the output affected? There is no change in the output.

16 The NOR Gate Summary The NOR gate produces a HIGH output only when both inputs are at the same logic level. The truth table is = Inputs Output The NOR operation shown as = +. lternatively, the NOR operation can be shown with a circled dot between the variables. Thus, it can be shown as =..

17 Summary The NOR Gate = Example waveforms: Notice that the NOR gate will produce a HIGH when both inputs are the same. This makes it useful for comparison functions. If the waveform is inverted but remains the same, how is the output affected? The output will be inverted.

18 Summary Fixed Function Logic Two major fixed function logic families are TTL and CMOS. third technology is icmos, which combines the first two. Packaging for fixed function logic is shown in in ±. in in Pin no. identifiers 4 Lead no. identifier DIP package SOIC package

19 Summary Fixed Function Logic Some common gate configurations are shown. V CC V CC V CC V CC ' GND ' GND ' GND ' GND V CC V CC V CC V CC ' GND ' GND ' GND ' GND V CC V CC V CC V CC ' GND ' GND ' GND ' GND

20 Fixed Function Logic Summary Logic symbols show the gates and associated pin numbers. () (2) (4) (5) (9) () (2) (3) V CC (4) (3) (6) (8) () () & (3) (2) (4) (5) (6) (9) () (8) (2) (3) () (7) GND

21 Fixed Function Logic Summary Data sheets include limits and conditions set by the manufacturer as well as DC and C characteristics. For example, some maximum ratings for a 74HC are: V CC MIMUM RTINGS Symbol Parameter Value Unit DC Supply Voltage (Referenced to GND).5 to + 7. V V V in DC Input Voltage (Referenced to GND).5 to V CC +.5 V V V out DC Output Voltage (Referenced to GND).5 to V CC +.5 V V I in DC Input Current, per pin ± 2 m DC Output Current, per pin ± 25 m I CC DC Supply Current, V CC and GND pins ± 5 m P D Power Dissipation in Still ir, Plastic or Ceramic DIP 75 mw T stg SOIC Package TSSOP Package Storage Temperature to + 5 C T L Lead Temperature, mm from Case for Seconds C Plastic DIP, SOIC, or TSSOP Package 26 Ceramic DIP 3

22 Digital Fundamentals Tenth Edition Floyd Chapter 4 Modified by Yuttapong Jiraraksopakun Floyd, Digital Fundamentals, th 28 Pearson Education ENE, KMUTT ed 29

23 Summary oolean ddition In oolean algebra, a variable is a symbol used to represent an action, a condition, or data. single variable can only have a value of or. The complement represents the inverse of a variable and is indicated with an overbar. Thus, the complement of is. literal is a variable or its complement. ddition 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. Determine the values of,, and C that make the sum term of the expression + + C =? Each literal must = ; therefore =, = and C =.

24 oolean Multiplication Summary In oolean algebra, multiplication is equivalent to the ND 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, and C if the product term of.. C =? Each literal must = ; therefore =, = and C =.

25 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. + = + For multiplication, the commutative law states In terms of the result, the order in which variables are NDed makes no difference. =

26 Summary ssociative 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. + ( +C) = ( + ) + C For multiplication, the associative law states When NDing more than two variables, the result is the same regardless of the grouping of the variables. (C) = ()C

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

28 Rules of oolean lgebra Summary. + = 2. + = 3.. = 4.. = 5. + = 6. + = 7.. = 8.. = = 9. =. + =. + = + 2. ( + )( + C) = + C

29 Rules of oolean lgebra Summary Rules of oolean algebra can be illustrated with Venn diagrams. The variable is shown as an area. The rule + = can be illustrated easily with a diagram. dd an overlapping area to represent the variable. The overlap region between and represents. = The diagram visually shows that + =. Other rules can be illustrated with the diagrams as well.

30 Summary Rules of oolean lgebra Illustrate the rule + = + with a Venn diagram. This time, is represented by the blue area and again by the red circle. The intersection represents. Notice that + = +

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

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

33 DeMorgan s Theorem DeMorgan s st Theorem Summary The complement of a product of variables is equal to the sum of the complemented variables. = + pplying DeMorgan s first theorem to gates: + Inputs Output + NND Negative-OR

34 Summary DeMorgan s Theorem DeMorgan s 2 nd Theorem The complement of a sum of variables is equal to the product of the complemented variables. + =. pplying DeMorgan s second theorem to gates: + Inputs Output + NOR Negative-ND

35 DeMorgan s Theorem Summary pply DeMorgan s theorem to remove the overbar covering both terms from the expression = 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 = = C. D. Deleting the double bar gives = C. D.

36 C D Summary oolean nalysis 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 oolean algebra. pply oolean algebra to derive the expression for. Write the expression for each gate: ( + ) C ( + ) = C ( + )+ D pplying DeMorgan s theorem and the distribution law: = C ( ) + D = C + D

37 Summary SOP and POS forms oolean 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. n expression is in SOP form when two or more product terms are summed as in the following examples: C + C + C D C D + E n expression is in POS form when two or more sum terms are multiplied as in the following examples: ( + )( + C) ( + + C)( + D) ( + )C

38 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 = + C to standard form. The first term does not include the variable C. Therefore, multiply it by the (C + C), which = : = (C + C) + C = C + C + C

39 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 (rule 8: =) and applying rule 2, which states that ( + )( + C) = + C. Convert = ( + )( + + 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. = ( + + C C)( + + C) = ( + + C )( + + C)( + + C)

40 Summary Converting Standard SOP to Standard POS C = ++C C = ++C C = ++C C = ++C SOP active-high logic POS active-low logic

41 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,, 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

42 Summary Gray code Karnaugh maps Cells are usually labeled using 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.

43 Summary Karnaugh maps lternatively, 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 C The cells are C and C. C C C C

44 Summary SOP Standard form in Karnaugh Map

45 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. 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 C. 4. The horizontal group is read. = C +

46 Karnaugh maps 4-variable map has an adjacent cell on each of its four boundaries as shown. CD CD CD CD Summary 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

47 Summary Karnaugh maps Group the s on the map and read the minimum logic. changes changes CD C changes across outer boundary C changes. 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 D. 4. The lower (green) group is read as D. = D +D

48 Summary Converting Standard SOP to Standard POS with Karnaugh SOP = C + + C POS = (+C) (+) (+C) SOP = C+ C+

49 Don t Care Conditions Summary

50 Don t Care Conditions Summary

51 Summary Five-Variable Karnaugh Maps

52 Homework 7 Chapter 3 (2, 2, 8, 45) Chapter 4 (, 3, 2, 36 a, 44)

Summary. Boolean Addition

Summary. Boolean Addition 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

More information

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

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)

More information

Lecture 7 Logic Simplification

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

More information

DKT 122/3 DIGITAL SYSTEM 1

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

More information

2/8/2017. SOP Form Gives Good Performance. ECE 120: Introduction to Computing. K-Maps Can Identify Single-Gate Functions

2/8/2017. SOP Form Gives Good Performance. ECE 120: Introduction to Computing. K-Maps Can Identify Single-Gate Functions University of Illinois at Urbana-Champaign Dept. of Electrical and Computer Engineering ECE 120: Introduction to Computing Two-Level Logic SOP Form Gives Good Performance s you know, one can use a K-map

More information

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

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

More information

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

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

More information

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:

More information

Chapter 2. Boolean Expressions:

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

More information

Digital Logic Design (3)

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

More information

At this point in our study of digital circuits, we have two methods for representing combinational logic: schematics and truth tables.

At this point in our study of digital circuits, we have two methods for representing combinational logic: schematics and truth tables. HPTER FIVE oolean lgebra 5.1 Need for oolean Expressions t this point in our study of digital circuits, we have two methods for representing combinational logic: schematics and truth tables. 0 0 0 1 0

More information

Bawar Abid Abdalla. Assistant Lecturer Software Engineering Department Koya University

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

More information

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

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

More information

Bawar Abid Abdalla. Assistant Lecturer Software Engineering Department Koya University

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

More information

IT 201 Digital System Design Module II Notes

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.

More information

Digital Fundamentals

Digital Fundamentals Digital Fundamentals Tenth Edition Floyd Chapter 1 Modified by Yuttapong Jiraraksopakun Floyd, Digital Fundamentals, 10 th 2008 Pearson Education ENE, KMUTT ed 2009 Analog Quantities Most natural quantities

More information

Introduction to Microprocessors and Digital Logic (ME262) Boolean Algebra and Logic Equations. Spring 2011

Introduction to Microprocessors and Digital Logic (ME262) Boolean Algebra and Logic Equations. Spring 2011 Introduction to Microprocessors and Digital (ME262) lgebra and Spring 2 Outline. lgebra 2. 3. Karnaugh Maps () 4. Two-variable 5. 6. 7. 2 lgebra s of Simplifying equations are defined in terms of inary

More information

Section 001. Read this before starting! You may use one sheet of scrap paper that you will turn in with your test.

Section 001. Read this before starting! You may use one sheet of scrap paper that you will turn in with your test. Points missed: Student's Name: Total score: /100 points East Tennessee State University Department of Computer and Information Sciences CSCI 2150 (Tarnoff) Computer Organization TEST 1 for Fall Semester,

More information

Boolean Algebra and Logic Gates

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

More information

EEE130 Digital Electronics I Lecture #4_1

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

More information

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 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

More information

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)

More information

Experiment 4 Boolean Functions Implementation

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.

More information

MODULE 5 - COMBINATIONAL LOGIC

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

More information

2.6 BOOLEAN FUNCTIONS

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

More information

Chapter 2 Combinational

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

More information

ELCT201: DIGITAL LOGIC DESIGN

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

More information

ELCT201: DIGITAL LOGIC DESIGN

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

More information

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

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

More information

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

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),

More information

Chap-2 Boolean Algebra

Chap-2 Boolean Algebra Chap-2 Boolean Algebra Contents: My name Outline: My position, contact Basic information theorem and postulate of Boolean Algebra. or project description Boolean Algebra. Canonical and Standard form. Digital

More information

Identity. p ^ T p p _ F p. Idempotency. p _ p p p ^ p p. Associativity. (p _ q) _ r p _ (q _ r) (p ^ q) ^ r p ^ (q ^ r) Absorption

Identity. p ^ T p p _ F p. Idempotency. p _ p p p ^ p p. Associativity. (p _ q) _ r p _ (q _ r) (p ^ q) ^ r p ^ (q ^ r) Absorption dam lank pring 27 E 3 Identity Pre-Lecture Problem Do it! Do it now! What are you waiting for?! Use Logical Equivalences to show!! $! & & $! p ^ T p p _ F p Domination p _ T T p ^ F F Idempotency p _ p

More information

Experiment 3: Logic Simplification

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

More information

Learning Objectives: Topic Karnaugh Maps. At the end of this topic you will be able to;

Learning Objectives: Topic Karnaugh Maps. At the end of this topic you will be able to; Topic.2.3 Karnaugh Maps Learning Objectives: t the end of this topic you will be able to; Draw a Karnaugh map for a logic system with up to four inputs and use it to minimise the number of gates required;

More information

Chapter 2 Combinational Logic Circuits

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

More information

Combinational Logic & Circuits

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

More information

PART 1. Simplification Using Boolean Algebra

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

More information

Combinational Circuits Digital Logic (Materials taken primarily from:

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

More information

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 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

More information

Ch. 5 : Boolean Algebra &

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

More information

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

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

More information

Chapter 2. Boolean Algebra and Logic Gates

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

More information

Logic Design (Part 2) Combinational Logic Circuits (Chapter 3)

Logic Design (Part 2) Combinational Logic Circuits (Chapter 3) Digital Logic Circuits Logic Design (Part ) Combinational Logic Circuits (Chapter 3) ² We saw how we can build the simple logic gates using transistors ² Use these gates as building blocks to build more

More information

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 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

More information

Functional Block: Decoders

Functional Block: Decoders University of Wisconsin - Madison EE/omp Sci 352 Digital Systems Fundamentals harles R. Kime Section 2 Fall 2 hapter 3 ombinational Logic Design Part 2 Tom Kaminski & harles R. Kime harles Kime & Thomas

More information

Chapter 2 Combinational Logic Circuits

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

More information

Announcements. Chapter 2 - Part 1 1

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

More information

9/10/2016. ECE 120: Introduction to Computing. The Domain of a Boolean Function is a Hypercube. List All Implicants for One Variable A

9/10/2016. ECE 120: Introduction to Computing. The Domain of a Boolean Function is a Hypercube. List All Implicants for One Variable A University of Illinois at Urbana-Champaign Dept. of Electrical and Computer Engineering ECE 120: Introduction to Computing To Simplify, Write Function as a Sum of Prime Implicants One way to simplify a

More information

Combinational Logic Circuits

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

More information

Unit-IV Boolean Algebra

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

More information

Boolean Algebra. BME208 Logic Circuits Yalçın İŞLER

Boolean Algebra. BME208 Logic Circuits Yalçın İŞLER Boolean Algebra BME28 Logic Circuits Yalçın İŞLER islerya@yahoo.com http://me.islerya.com 5 Boolean Algebra /2 A set of elements B There exist at least two elements x, y B s. t. x y Binary operators: +

More information

Lecture (05) Boolean Algebra and Logic Gates

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

More information

1. Mark the correct statement(s)

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

More information

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

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?

More information

University of Technology

University of Technology University of Technology Lecturer: Dr. Sinan Majid Course Title: microprocessors 4 th year Lecture 5 & 6 Minimization with Karnaugh Maps Karnaugh maps lternate way of representing oolean function ll rows

More information

Digital Techniques. Lecture 1. 1 st Class

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

More information

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

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

More information

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

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

More information

Gate-Level Minimization. BME208 Logic Circuits Yalçın İŞLER

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

More information

CS8803: Advanced Digital Design for Embedded Hardware

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

More information

CprE 281: Digital Logic

CprE 281: Digital Logic CprE 28: Digital Logic Instructor: Alexander Stoytchev http://www.ece.iastate.edu/~alexs/classes/ Minimization CprE 28: Digital Logic Iowa State University, Ames, IA Copyright Alexander Stoytchev Administrative

More information

Objectives: 1. Design procedure. 2. Fundamental circuits. 1. Design procedure

Objectives: 1. Design procedure. 2. Fundamental circuits. 1. Design procedure Objectives: 1. Design procedure. 2. undamental circuits. 1. Design procedure Design procedure has five steps: o Specification. o ormulation. o Optimization. o Technology mapping. o Verification. Specification:

More information

Chapter 3. Gate-Level Minimization. Outlines

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

More information

Boolean relationships on Venn Diagrams

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

More information

LECTURE 4. Logic Design

LECTURE 4. Logic Design LECTURE 4 Logic Design LOGIC DESIGN The language of the machine is binary that is, sequences of 1 s and 0 s. But why? At the hardware level, computers are streams of signals. These signals only have two

More information

Boolean logic. Boolean Algebra. Introduction to Computer Yung-Yu Chuang NOT AND NOT

Boolean logic. Boolean Algebra. Introduction to Computer Yung-Yu Chuang NOT AND NOT oolean lgebra oolean logic ased on symbolic logic, designed by George oole oolean variables take values as or. oolean expressions created from: NOT, ND, OR Introduction to Computer ung-u Chuang with slides

More information

Digital Circuits ECS 371

Digital Circuits ECS 371 Digital Circuits ECS 37 Dr. Prapun Suksompong prapun@siit.tu.ac.th Lecture 7 Office Hours: KD 36-7 Monday 9:-:3, :3-3:3 Tuesday :3-:3 Announcement HW2 posted on the course web site Chapter 4: Write down

More information

Incompletely Specified Functions with Don t Cares 2-Level Transformation Review Boolean Cube Karnaugh-Map Representation and Methods Examples

Incompletely Specified Functions with Don t Cares 2-Level Transformation Review Boolean Cube Karnaugh-Map Representation and Methods Examples Lecture B: Logic Minimization Incompletely Specified Functions with Don t Cares 2-Level Transformation Review Boolean Cube Karnaugh-Map Representation and Methods Examples Incompletely specified functions

More information

Combinational Logic Circuits Part III -Theoretical Foundations

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

More information

Gate Level Minimization Map Method

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

More information

Midterm Exam Review. CS 2420 :: Fall 2016 Molly O'Neil

Midterm Exam Review. CS 2420 :: Fall 2016 Molly O'Neil Midterm Exam Review CS 2420 :: Fall 2016 Molly O'Neil Midterm Exam Thursday, October 20 In class, pencil & paper exam Closed book, closed notes, no cell phones or calculators, clean desk 20% of your final

More information

Module -7. Karnaugh Maps

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)

More information

X Y Z F=X+Y+Z

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

More information

UNIT 2 BOOLEAN ALGEBRA

UNIT 2 BOOLEAN ALGEBRA UNIT 2 BOOLEN LGEBR Spring 2 2 Contents Introduction Basic operations Boolean expressions and truth tables Theorems and laws Basic theorems Commutative, associative, and distributive laws Simplification

More information

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

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,

More information

There are only 16 possible 2-input gates Let s examine all of them. Some we already know, others are just silly.

There are only 16 possible 2-input gates Let s examine all of them. Some we already know, others are just silly. ll the Gates There are only 6 possible 2-input gates Let s examine all of them. Some we already know, others are just silly. Do we really need all of these gates? How many of these gates can be implemented

More information

Lecture 6: Signed Numbers & Arithmetic Circuits. BCD (Binary Coded Decimal) Points Addressed in this Lecture

Lecture 6: Signed Numbers & Arithmetic Circuits. BCD (Binary Coded Decimal) Points Addressed in this Lecture Points ddressed in this Lecture Lecture 6: Signed Numbers rithmetic Circuits Professor Peter Cheung Department of EEE, Imperial College London (Floyd 2.5-2.7, 6.1-6.7) (Tocci 6.1-6.11, 9.1-9.2, 9.4) Representing

More information

Homework. Update on website issue Reading: Chapter 7 Homework: All exercises at end of Chapter 7 Due 9/26

Homework. Update on website issue Reading: Chapter 7 Homework: All exercises at end of Chapter 7 Due 9/26 Homework Update on website issue Reading: hapter 7 Homework: All exercises at end of hapter 7 Due 9/26 opyright c 22 28 UMaine omputer Science Department / 2 OS 4: Foundations of omputer Science Karnaugh

More information

Dr. Chuck Cartledge. 10 June 2015

Dr. Chuck Cartledge. 10 June 2015 Miscellanea Exam #1 Break Exam review 2.1 2.2 2.3 2.4 Break 3 4 Conclusion References CSC-205 Computer Organization Lecture #003 Chapter 2, Sections 2.1 through 4 Dr. Chuck Cartledge 10 June 2015 1/30

More information

ENGIN 112 Intro to Electrical and Computer Engineering

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

More information

Specifying logic functions

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

More information

LAB #1 BASIC DIGITAL CIRCUIT

LAB #1 BASIC DIGITAL CIRCUIT LAB #1 BASIC DIGITAL CIRCUIT OBJECTIVES 1. To study the operation of basic logic gates. 2. To build a logic circuit from Boolean expressions. 3. To introduce some basic concepts and laboratory techniques

More information

Gate-Level Minimization. section instructor: Ufuk Çelikcan

Gate-Level Minimization. section instructor: Ufuk Çelikcan Gate-Level Minimization section instructor: Ufuk Çelikcan Compleity of Digital Circuits Directly related to the compleity of the algebraic epression we use to build the circuit. Truth table may lead to

More information

Lecture (04) Boolean Algebra and Logic Gates

Lecture (04) Boolean Algebra and Logic Gates Lecture (4) Boolean Algebra and Logic Gates By: Dr. Ahmed ElShafee ١ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Boolean algebra properties basic assumptions and properties: Closure law A set S is

More information

Lecture (04) Boolean Algebra and Logic Gates By: Dr. Ahmed ElShafee

Lecture (04) Boolean Algebra and Logic Gates By: Dr. Ahmed ElShafee Lecture (4) Boolean Algebra and Logic Gates By: Dr. Ahmed ElShafee Boolean algebra properties basic assumptions and properties: Closure law A set S is closed with respect to a binary operator, for every

More information

Combinational Logic Circuits

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

More information

Presented By :- Alok Kumar Lecturer in ECE C.R.Polytechnic, Rohtak

Presented By :- Alok Kumar Lecturer in ECE C.R.Polytechnic, Rohtak Presented By :- Alok Kumar Lecturer in ECE C.R.Polytechnic, Rohtak Content - Introduction -2 Feature -3 Feature of BJT -4 TTL -5 MOS -6 CMOS -7 K- Map - Introduction Logic IC ASIC: Application Specific

More information

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

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

More information

Binary logic. Dr.Abu-Arqoub

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

More information

Unit 6 1.Random Access Memory (RAM) Chapter 3 Combinational Logic Design 2.Programmable Logic

Unit 6 1.Random Access Memory (RAM) Chapter 3 Combinational Logic Design 2.Programmable Logic EE 200: Digital Logic Circuit Design Dr Radwan E Abdel-Aal, COE Unit 6.Random Access Memory (RAM) Chapter 3 Combinational Logic Design 2. Logic Logic and Computer Design Fundamentals Part Implementation

More information

Digital Logic Lecture 7 Gate Level Minimization

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

More information

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

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

More information

DIGITAL CIRCUIT LOGIC UNIT 7: MULTI-LEVEL GATE CIRCUITS NAND AND NOR GATES

DIGITAL CIRCUIT LOGIC UNIT 7: MULTI-LEVEL GATE CIRCUITS NAND AND NOR GATES DIGITAL CIRCUIT LOGIC UNIT 7: MULTI-LEVEL GATE CIRCUITS NAND AND NOR GATES 1 iclicker Question 13 Considering the K-Map, f can be simplified as (2 minutes): A) f = b c + a b c B) f = ab d + a b d AB CD

More information

Read this before starting!

Read this before starting! Points missed: Student's Name: Total score: /100 points East Tennessee State University Department of Computer and Information Sciences CSCI 2150 (Tarnoff) Computer Organization TEST 1 for Spring Semester,

More information

Simplification of two-level combinational logic

Simplification of two-level combinational logic ombinational logic optimization! lternate representations of oolean functions " cubes " karnaugh maps! Simplification " two-level simplification " exploiting don t cares " algorithm for simplification

More information

Principles of Digital Techniques PDT (17320) Assignment No State advantages of digital system over analog system.

Principles of Digital Techniques PDT (17320) Assignment No State advantages of digital system over analog system. Assignment No. 1 1. State advantages of digital system over analog system. 2. Convert following numbers a. (138.56) 10 = (?) 2 = (?) 8 = (?) 16 b. (1110011.011) 2 = (?) 10 = (?) 8 = (?) 16 c. (3004.06)

More information

Section 001. Read this before starting!

Section 001. Read this before starting! Points missed: Student's Name: Total score: / points East Tennessee State University Department of Computer and Information Sciences CSCI 25 (Tarnoff) Computer Organization TEST 2 for Fall Semester, 25

More information

Standard Forms of Expression. Minterms and Maxterms

Standard Forms of Expression. Minterms and Maxterms Standard Forms of Expression Minterms and Maxterms Standard forms of expressions We can write expressions in many ways, but some ways are more useful than others A sum of products (SOP) expression contains:

More information

A graphical method of simplifying logic

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

More information

Lecture 4: Implementation AND, OR, NOT Gates and Complement

Lecture 4: Implementation AND, OR, NOT Gates and Complement EE210: Switching Systems Lecture 4: Implementation AND, OR, NOT Gates and Complement Prof. YingLi Tian Feb. 13, 2018 Department of Electrical Engineering The City College of New York The City University

More information