# Logic Design: Part 2

Size: px
Start display at page:

Transcription

1 Orange Coast College Business Division Computer Science Department CS 6- Computer Architecture Logic Design: Part 2

2 Where are we? Number systems Decimal Binary (and related Octal and Hexadecimal) Binary encodings BCD (842 and XS3) Gray ASCII, EBCDIC, Unicode OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 2 2

3 What's next? Binary Arithmetic Addition (nefarious) and Subtraction Multiplication and Division Boolean Algebra Not your mom's high school algebra OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 3 3

4 Binary Addition Simply adding two single digit numbers Only 4 possible combinations A B Sum Carry OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 4 4

5 Binary Addition What happens when we consider a carry-in Essentially adding three single digits 8 possible combinations Carry-In A B Sum Carry-Out OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 5 5

6 Binary Addition- Bring it together When two 's are added together, their sum is in base 2 For the LSBs of the numbers to be added, there are only two digits to consider For all the following digits, they may have a carry digit from the previous column As we see later, this can be expressed as a single boolean function OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 6 6

7 Binary Subtraction Subtracting two digits 4 possible combinations A B Difference Borrow OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 7 7

8 Binary Subtraction Subtracting 3 digits Considering the borrow-in Self-exercise Borrow-In A B Difference Borrow OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 8 8

9 Is there a better way? Subtraction with borrow is not efficient for digital systems Why? Trust me it is. Hint: Extra logic required What is the better way? Complements OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 9 9

10 Subtraction through Complements Problem: Find X-Y X = (2) = () 2 Y = (9) = () 2 Step : Convert Y to 2's complement form 2's complement of Y = () 2 Step 2: Add the result to X + = (2) OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty)

11 Binary Multiplication The product of two binary numbers is Only if both numbers are The product of two binary numbers is If any of the number is Size of result is the combined size of the operands Details Can be done as a group of shifting and additions. Can also be carried out with fractional numbers. OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty)

12 Binary Multiplication: Example: (27) = (2) = x = = (567) OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 2 2

13 Binary Division Similar to decimal number divisions. Converted to a group of shifts, subtractions, & comparisons. Can also be applied to fractional numbers. Causes overflow if the divisor is zero or the quotient needs more bits to express than the computer word (or register) size. There are special techniques for performing division directly on 2's complement numbers. OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 3 3

14 Binary Division Example (5) = (5) = (5) / (5) = () quotient dividend shifted divisor reduced dividend shifted divisor remainder OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 4 4

15 Boolean Algebra The algebra of logic Express logic functions with logic equations. Used in electronic circuits to build computers and other logic circuits Analyze and describe the behavior of logic circuits Amaze and astound your friends OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 5 5

16 Hmm, what is an algebra? A formal mathematical system Consists of a set of objects Operations on those objects. Examples Numerical algebra Set algebra Matrix algebra Boolean algebra OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 6

17 Boolean Algebra Variables Values Operations AND, OR, NOT Laws a + = A + = Not(Not(a) = a... OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 7 7

18 Variables and Values Variables Named a, b, c,.. Should be of Boolean type Represent a changing value i.e. Can represent any value from the input domain Values Domain: and (True & False) OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 8 8

19 Operators Not Negation/Inversion Symbols: ',! (in C), A Unary Or Logical Sum Symbols: +, (in C) And Logical Product Symbols: *, && (in C) OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 9

20 Operator Precedence Determines the order to evaluate From highest to lowest ( )- Parentheses Not And Or What does this mean? Consider: A+'B*C Negate B, And the result with C Or with A OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 2

21 Boolean Functions Formula that has variables and operators The 'output' value is on the left hand side The variables and operators on the right Examples: X = A * B ( A AND B) X is true if and only if: Both A and B are true X = A + B * 'C (A OR B AND NOT C) X is true if and only if: Either A is true or B is true and C is false OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 2 2

22 A bit more formal... Literal A variable or the complement of the variable A, B or A', B' Product Term A Single literals AND of two or more literals Sum Term A single literal OR of two or more literals OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 22

23 Sample Terms Product Term A' A * B' * C Sum Term B A + B + D OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 23

24 Combining the previous Product of sums Logical AND of Sum Terms A' * (A + B + C) * (A + B' + C) * (A' + B' + C) Sum of products Logical OR of Product Terms A' + (A * B * C) + (A * B' * C) + (A' * B' * C) OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 24

25 More formal descriptions Normal term An expression where no variable appears more than once N-Variable Minterm A Normal Product Term of N-Literals Every variable appears exactly once N-Variable Maxterm A Normal Sum Term of N-Literals Every variable appears exactly once OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 25

26 Lets look at an example 4-variable Minterm s: A B C D A B' C D A B C' D A' B' C' D' 4-variable Maxterm s: A + B + C+ D A' + B' + C+ D A + B + C'+ D A' + B' + C'+ D' OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 26

27 What can we do with these? We have all elements of an algebra Values Variables Operators Use these elements to derive a series of laws Laws are useful because Allow us to find equivalent functions May provide for more efficient circuits Simplify a given function May provide for cheaper circuits OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 27 27

28 Laws of Boolean Algebra Identity A + = A A * = A Dominance A + = A * = Idempotent A * A = A A + A = A OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 28

29 Laws of Boolean Algebra Inverse (Complement) A + A' = A * A' = Involution (Law of the double complement) (A') ' = A Two negatives actually make a positive! OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 29

30 Laws of Boolean Algebra Commutative A + B = B + A A * B = B * A Associative A + (B + C) = (A + B) + C A * (B * C) = (A * B) * C Distributive A * (B + C) = (A * B) + (A * C) A + (B * C) = (A+B) * (A+C) OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 3

31 Laws of Boolean Algebra Covering (Absorption/Redundance) A + A*B = A A * ( A + B ) = A A + A' * B = A + B A * (A' + B) = A * B Combining A * B + A * B' = A (A + B) * (A + B' ) = A Consensus A * B + A' * C + B * C = A * B + A' * C (A + B) * (A' + C) * (B + C) = (A + B) * (A' + C) OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 3 3

32 Laws of Boolean Algebra Special set of laws DeMorgan's Laws DeMorgan's states (A + B)' = A' B' (NOR) (A B)' = A' + B' (NAND) DeMorgan's theorems apply to more than 2 variables OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 32

33 Examples Applying DeMorgan's X = ((A' + C) * (B + D'))' = (A' + C) ' + (B + D') ' By DeMorgan's = (A ' ' * C ' ) + (B ' * D ' ') By DeMorgan's = A * C ' + B ' * D Another example X = A' * B * C + A' * B * C' = A' * (B * C + B * C') By Distributive = A' * (B * ( C + C')) By Distributive = A' * (B * ) By Inverse = A' * B By Identity OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 33

34 Interesting tidbits... Duality principle: All laws of Boolean algebra are stated in pairs They are obtained by: swapping 's and 's, and swapping +'s with *'s OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 34

35 More tidbits... The correctness of the laws can be proven by Building their equivalent logic circuits & checking Building the truth table of the function Using Venn diagrams Suitable for up to 3 variables Derivation The successive application of other laws OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 35

36 Truth Table Completely describe any combinational logic function Describe the output with respect to the input Each entry specifies the value of the output for that particular input combination Each output column represents a different function Functions may be directly constructed from the truth table For a function with n-inputs, there are 2 n entries in the truth table OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 36

37 OCC - CS/CIS CS6-Ch-Orientation Morgan Kaufmann Publishers ( Augmented & Modified by M.Malaty) 37 OCC-CS 6 Fall Morgan Kaufmann Publishers (Augmented & Modified by M.Malaty and M. Beers) Truth Table Examples Outputs D E F Inputs A B C

38 Convert a Truth Table How do we convert a truth table to (one of) its associated function(s)? For each place where output is Create an AND expression of the inputs If input is, negate its value Else, just use the variable name OR all the expressions together OCC-CS 6 Fall 23 A B Output(C) C = AB + AB OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 38

39 Now, what does a truth table look like? Hint, think addition Inputs A B Outputs Sum Carry OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 39

40 Surprise, In-class exercise Join a different group than you were in before Goal: Create the boolean expression for each of the following Simple single digit binary addition table Sum Carry Binary addition with carry table Sum Carry Simple subtraction Difference Borrow OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 4

41 Some Help ) Simple Addition 3) Simple Subtraction A B Sum Carry A B Diff Borrow 2) Addition with Carry-in Carry-In A B Sum Carry-Out OCC-CS 6 Fall 23 OCC - CS/CIS CS6-Ch-Orientation 998 Morgan Kaufmann Publishers (Augmented 998 Morgan & Kaufmann Modified Publishers by M.Malaty ( Augmented and M. & Modified Beers) by M.Malaty) 4 4

### 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

### 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

### 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

### Propositional Calculus: Boolean Algebra and Simplification. CS 270: Mathematical Foundations of Computer Science Jeremy Johnson

Propositional Calculus: Boolean Algebra and Simplification CS 270: Mathematical Foundations of Computer Science Jeremy Johnson Propositional Calculus Topics Motivation: Simplifying Conditional Expressions

### 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

### CS February 17

Discrete Mathematics CS 26 February 7 Equal Boolean Functions Two Boolean functions F and G of degree n are equal iff for all (x n,..x n ) B, F (x,..x n ) = G (x,..x n ) Example: F(x,y,z) = x(y+z), G(x,y,z)

### BINARY SYSTEM. Binary system is used in digital systems because it is:

CHAPTER 2 CHAPTER CONTENTS 2.1 Binary System 2.2 Binary Arithmetic Operation 2.3 Signed & Unsigned Numbers 2.4 Arithmetic Operations of Signed Numbers 2.5 Hexadecimal Number System 2.6 Octal Number System

### 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 Fall Semester,

### 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

### EE292: Fundamentals of ECE

EE292: Fundamentals of ECE Fall 2012 TTh 10:00-11:15 SEB 1242 Lecture 22 121115 http://www.ee.unlv.edu/~b1morris/ee292/ 2 Outline Review Binary Number Representation Binary Arithmetic Combinatorial Logic

### 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

### 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 Data Representation Binary Codes Why 6-3-1-1 and Excess-3? Basic Operations of Boolean Algebra Examples

### 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

### 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

### 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

### 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

### 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

### CHW 261: Logic Design

CHW 261: Logic Design Instructors: Prof. Hala Zayed Dr. Ahmed Shalaby http://www.bu.edu.eg/staff/halazayed14 http://bu.edu.eg/staff/ahmedshalaby14# Slide 1 Slide 2 Slide 3 Digital Fundamentals CHAPTER

### 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

### FUNDAMENTALS OF DIGITAL CIRCUITS

FUNDAMENTALS OF DIGITAL CIRCUITS THIRD EDITION A. Anand Kumar Principal K.L. University College of Engineering K.L. University Green Fields, Vaddeswaram Guntur District Andhra Pradesh Delhi-110092 2014

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)

### CS/COE 0447 Example Problems for Exam 2 Spring 2011

CS/COE 0447 Example Problems for Exam 2 Spring 2011 1) Show the steps to multiply the 4-bit numbers 3 and 5 with the fast shift-add multipler. Use the table below. List the multiplicand (M) and product

### Chapter 6 Primitive types

Chapter 6 Primitive types Lesson page 6-1. Primitive types Question 1. There are an infinite number of integers, so it would be too ineffient to have a type integer that would contain all of them. Question

### 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.

### Digital Fundamentals

Digital Fundamentals Tenth Edition Floyd Chapter 2 2009 Pearson Education, Upper 2008 Pearson Saddle River, Education NJ 07458. All Rights Reserved Decimal Numbers The position of each digit in a weighted

### 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

### Propositional Calculus. CS 270: Mathematical Foundations of Computer Science Jeremy Johnson

Propositional Calculus CS 270: Mathematical Foundations of Computer Science Jeremy Johnson Propositional Calculus Objective: To provide students with the concepts and techniques from propositional calculus

### Arithmetic and Bitwise Operations on Binary Data

Arithmetic and Bitwise Operations on Binary Data CSCI 2400: Computer Architecture ECE 3217: Computer Architecture and Organization Instructor: David Ferry Slides adapted from Bryant & O Hallaron s slides

### CPS 104 Computer Organization and Programming

CPS 104 Computer Organization and Programming Lecture 9: Integer Arithmetic. Robert Wagner CPS104 IMD.1 RW Fall 2000 Overview of Today s Lecture: Integer Multiplication and Division. Read Appendix B CPS104

### 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,

### Summary of Course Coverage

CS-227, Discrete Structures I Spring 2006 Semester Summary of Course Coverage 1) Propositional Calculus a) Negation (logical NOT) b) Conjunction (logical AND) c) Disjunction (logical inclusive-or) d) Inequalities

### COPYRIGHTED MATERIAL INDEX

INDEX Absorption law, 31, 38 Acyclic graph, 35 tree, 36 Addition operators, in VHDL (VHSIC hardware description language), 192 Algebraic division, 105 AND gate, 48 49 Antisymmetric, 34 Applicable input

Class Subject Code Subject Prepared By Lesson Plan for Time: Lesson. No 1.CONTENT LIST: Introduction to UnitI 2. SKILLS ADDRESSED: Listening I year, 02 sem CS6201 Digital Principles & System Design S.Seedhanadevi

### 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)

### MACHINE LEVEL REPRESENTATION OF DATA

MACHINE LEVEL REPRESENTATION OF DATA CHAPTER 2 1 Objectives Understand how integers and fractional numbers are represented in binary Explore the relationship between decimal number system and number systems

### 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

### 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.

### Lecture 2: Number Systems

Lecture 2: Number Systems Syed M. Mahmud, Ph.D ECE Department Wayne State University Original Source: Prof. Russell Tessier of University of Massachusetts Aby George of Wayne State University Contents

### UNCA CSCI 255 Exam 1 Spring February, This is a closed book and closed notes exam. It is to be turned in by 1:45 PM.

UNCA CSCI 255 Exam 1 Spring 2017 27 February, 2017 This is a closed book and closed notes exam. It is to be turned in by 1:45 PM. Communication with anyone other than the instructor is not allowed during

### 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

### 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

### Chapter 5: Computer Arithmetic. In this chapter you will learn about:

Slide 1/29 Learning Objectives In this chapter you will learn about: Reasons for using binary instead of decimal numbers Basic arithmetic operations using binary numbers Addition (+) Subtraction (-) Multiplication

### Digital Fundamentals. CHAPTER 2 Number Systems, Operations, and Codes

Digital Fundamentals CHAPTER 2 Number Systems, Operations, and Codes Decimal Numbers The decimal number system has ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 The decimal numbering system has a base of

### DLD VIDYA SAGAR P. potharajuvidyasagar.wordpress.com. Vignana Bharathi Institute of Technology UNIT 1 DLD P VIDYA SAGAR

UNIT I Digital Systems: Binary Numbers, Octal, Hexa Decimal and other base numbers, Number base conversions, complements, signed binary numbers, Floating point number representation, binary codes, error

### 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,

### [Ch 6] Set Theory. 1. Basic Concepts and Definitions. 400 lecture note #4. 1) Basics

400 lecture note #4 [Ch 6] Set Theory 1. Basic Concepts and Definitions 1) Basics Element: ; A is a set consisting of elements x which is in a/another set S such that P(x) is true. Empty set: notated {

### 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

### Computer Organization and Programming

Sep 2006 Prof. Antônio Augusto Fröhlich (http://www.lisha.ufsc.br) 8 Computer Organization and Programming Prof. Dr. Antônio Augusto Fröhlich guto@lisha.ufsc.br http://www.lisha.ufsc.br/~guto Sep 2006

### ELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT, OAKLAND UNIVERSITY ECE-2700: Digital Logic Design Winter Notes - Unit 4. hundreds.

UNSIGNED INTEGER NUMBERS Notes - Unit 4 DECIMAL NUMBER SYSTEM A decimal digit can take values from to 9: Digit-by-digit representation of a positive integer number (powers of ): DIGIT 3 4 5 6 7 8 9 Number:

### 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

### Learning Objectives. Binary over Decimal. In this chapter you will learn about:

Ref Page Slide 1/29 Learning Objectives In this chapter you will learn about: Reasons for using binary instead of decimal numbers Basic arithmetic operations using binary numbers Addition (+) Subtraction

### Numbering systems. Dr Abu Arqoub

Numbering systems The decimal numbering system is widely used, because the people Accustomed (معتاد) to use the hand fingers in their counting. But with the development of the computer science another

### Review. EECS Components and Design Techniques for Digital Systems. Lec 05 Boolean Logic 9/4-04. Seq. Circuit Behavior. Outline.

Review EECS 150 - Components and Design Techniques for Digital Systems Lec 05 Boolean Logic 94-04 David Culler Electrical Engineering and Computer Sciences University of California, Berkeley Design flow

### Digital logic fundamentals. Question Bank. Unit I

Digital logic fundamentals Question Bank Subject Name : Digital Logic Fundamentals Subject code: CA102T Staff Name: R.Roseline Unit I 1. What is Number system? 2. Define binary logic. 3. Show how negative

### Binary Values. CSE 410 Lecture 02

Binary Values CSE 410 Lecture 02 Lecture Outline Binary Decimal, Binary, and Hexadecimal Integers Why Place Value Representation Boolean Algebra 2 First: Why Binary? Electronic implementation Easy to store

### 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

### 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.

### 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

### 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

### 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

### 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

### Moodle WILLINGDON COLLEGE SANGLI. ELECTRONICS (B. Sc.-I) Introduction to Number System

Moodle 1 WILLINGDON COLLEGE SANGLI ELECTRONICS (B. Sc.-I) Introduction to Number System E L E C T R O N I C S Introduction to Number System and Codes Moodle developed By Dr. S. R. Kumbhar Department of

### ELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT, OAKLAND UNIVERSITY ECE-278: Digital Logic Design Fall Notes - Unit 4. hundreds.

ECE-78: Digital Logic Design Fall 6 UNSIGNED INTEGER NUMBERS Notes - Unit 4 DECIMAL NUMBER SYSTEM A decimal digit can take values from to 9: Digit-by-digit representation of a positive integer number (powers

### Computer Logical Organization Tutorial

Computer Logical Organization Tutorial COMPUTER LOGICAL ORGANIZATION TUTORIAL Simply Easy Learning by tutorialspoint.com tutorialspoint.com i ABOUT THE TUTORIAL Computer Logical Organization Tutorial Computer

### 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

### Read this before starting!

Points missed: Student's Name: Total score: /1 points East Tennessee State University Department of Computer and Information Sciences CSCI 215 (Tarnoff) Computer Organization Section 1 TEST 1 for Fall

### Number Systems and Conversions UNIT 1 NUMBER SYSTEMS & CONVERSIONS. Number Systems (2/2) Number Systems (1/2) Iris Hui-Ru Jiang Spring 2010

Contents Number systems and conversion Binary arithmetic Representation of negative numbers Addition of two s complement numbers Addition of one s complement numbers Binary s Readings Unit.~. UNIT NUMBER

### Microcomputers. Outline. Number Systems and Digital Logic Review

Microcomputers Number Systems and Digital Logic Review Lecture 1-1 Outline Number systems and formats Common number systems Base Conversion Integer representation Signed integer representation Binary coded

### D I G I T A L C I R C U I T S E E

D I G I T A L C I R C U I T S E E Digital Circuits Basic Scope and Introduction This book covers theory solved examples and previous year gate question for following topics: Number system, Boolean algebra,

### Agenda EE 224: INTRODUCTION TO DIGITAL CIRCUITS & COMPUTER DESIGN. Lecture 1: Introduction. Go over the syllabus 3/31/2010

// EE : INTRODUCTION TO DIGITAL CIRCUITS & COMPUTER DESIGN Lecture : Introduction /9/ Avinash Kodi, kodi@ohio.edu Agenda Go over the syllabus Introduction ti to Digital it Systems // Why Digital Systems?

### Code No: 07A3EC03 Set No. 1

Code No: 07A3EC03 Set No. 1 II B.Tech I Semester Regular Examinations, November 2008 SWITCHING THEORY AND LOGIC DESIGN ( Common to Electrical & Electronic Engineering, Electronics & Instrumentation Engineering,

### ECEN 468 Advanced Logic Design

ECEN 468 Advanced Logic Design Lecture 26: Verilog Operators ECEN 468 Lecture 26 Operators Operator Number of Operands Result Arithmetic 2 Binary word Bitwise 2 Binary word Reduction 1 Bit Logical 2 Boolean

### 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

### KNOWLEDGE OF NUMBER SENSE, CONCEPTS, AND OPERATIONS

DOMAIN I. COMPETENCY 1.0 MATHEMATICS KNOWLEDGE OF NUMBER SENSE, CONCEPTS, AND OPERATIONS Skill 1.1 Compare the relative value of real numbers (e.g., integers, fractions, decimals, percents, irrational

### Introduction to Boolean logic and Logical Gates

Introduction to Boolean logic and Logical Gates Institute of Statistics Fall 2014 We saw the importance of the binary number system for data representation in a computer system. We ll see that the construction

### Introduction to Boolean Algebra

Introduction to Boolean Algebra Boolean algebra which deals with two-valued (true / false or and ) variables and functions find its use in modern digital computers since they too use two-level systems

### DIGITAL SYSTEM FUNDAMENTALS (ECE 421) DIGITAL ELECTRONICS FUNDAMENTAL (ECE 422) COURSE / CODE NUMBER SYSTEM

COURSE / CODE DIGITAL SYSTEM FUNDAMENTALS (ECE 421) DIGITAL ELECTRONICS FUNDAMENTAL (ECE 422) NUMBER SYSTEM A considerable subset of digital systems deals with arithmetic operations. To understand the

### Definitions. 03 Logic networks Boolean algebra. Boolean set: B 0,

3. Boolean algebra 3 Logic networks 3. Boolean algebra Definitions Boolean functions Properties Canonical forms Synthesis and minimization alessandro bogliolo isti information science and technology institute

### that system. weighted value associated with it. numbers. a number. the absence of a signal. MECH 1500 Quiz 2 Review Name: Class: Date:

Name: Class: Date: MECH 1500 Quiz 2 Review True/False Indicate whether the statement is true or false. 1. The decimal system uses the number 9 as its base. 2. All digital computing devices perform operations

### Chapter 5: Computer Arithmetic

Slide 1/29 Learning Objectives Computer Fundamentals: Pradeep K. Sinha & Priti Sinha In this chapter you will learn about: Reasons for using binary instead of decimal numbers Basic arithmetic operations

### Introduction to Boolean Algebra

Introduction to Boolean Algebra Boolean algebra which deals with two-valued (true / false or and ) variables and functions find its use in modern digital computers since they too use two-level systems

### CHAPTER 2 (b) : AND CODES

DKT 122 / 3 DIGITAL SYSTEMS 1 CHAPTER 2 (b) : NUMBER SYSTEMS OPERATION AND CODES m.rizal@unimap.edu.my sitizarina@unimap.edu.my DECIMAL VALUE OF SIGNED NUMBERS SIGN-MAGNITUDE: Decimal values of +ve & -ve

### 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

### Learning Log Title: CHAPTER 3: ARITHMETIC PROPERTIES. Date: Lesson: Chapter 3: Arithmetic Properties

Chapter 3: Arithmetic Properties CHAPTER 3: ARITHMETIC PROPERTIES Date: Lesson: Learning Log Title: Date: Lesson: Learning Log Title: Chapter 3: Arithmetic Properties Date: Lesson: Learning Log Title:

### Operators & Expressions

Operators & Expressions Operator An operator is a symbol used to indicate a specific operation on variables in a program. Example : symbol + is an add operator that adds two data items called operands.

### 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

### Computer Arithmetic. Appendix A Fall 2003 Lec.03-58

Computer Arithmetic Appendix A 18-347 Fall 2003 Lec.03-58 Intro Computer Arithmetic Computers use the binary system Easy to implement in electronics: 1 is 1V, 0 is 0V Easy to implement with switches (transistors!)

### 9/10/10. Arithmetic Operators. Today. Assigning floats to ints. Arithmetic Operators & Expressions. What do you think is the output?

Arithmetic Operators Section 2.15 & 3.2 p 60-63, 81-89 1 Today Arithmetic Operators & Expressions o Computation o Precedence o Associativity o Algebra vs C++ o Exponents 2 Assigning floats to ints int

### Slide Set 1. for ENEL 339 Fall 2014 Lecture Section 02. Steve Norman, PhD, PEng

Slide Set 1 for ENEL 339 Fall 2014 Lecture Section 02 Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary Fall Term, 2014 ENEL 353 F14 Section

### Number Systems CHAPTER Positional Number Systems

CHAPTER 2 Number Systems Inside computers, information is encoded as patterns of bits because it is easy to construct electronic circuits that exhibit the two alternative states, 0 and 1. The meaning of

### 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

### 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,

### Lecture (03) Binary Codes Registers and Logic Gates

Lecture (03) Binary Codes Registers and Logic Gates By: Dr. Ahmed ElShafee Binary Codes Digital systems use signals that have two distinct values and circuit elements that have two stable states. binary

### 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,

### GO - OPERATORS. This tutorial will explain the arithmetic, relational, logical, bitwise, assignment and other operators one by one.

http://www.tutorialspoint.com/go/go_operators.htm GO - OPERATORS Copyright tutorialspoint.com An operator is a symbol that tells the compiler to perform specific mathematical or logical manipulations.

### Electronic Data and Instructions

Lecture 2 - The information Layer Binary Values and Number Systems, Data Representation. Know the different types of numbers Describe positional notation Convert numbers in other bases to base 10 Convert

### Programming Lecture 3

Programming Lecture 3 Expressions (Chapter 3) Primitive types Aside: Context Free Grammars Constants, variables Identifiers Variable declarations Arithmetic expressions Operator precedence Assignment statements

### 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