Circuit analysis summary

Size: px
Start display at page:

Transcription

1 Boolean Algebra

2 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 between expressions and truth tables. Find the circuit s inputs and outputs Find a Boolean expression for the circuit Find a truth table for the circuit CS231 Boolean Algebra 2

3 Boolean Functions summary We can interpret high or low voltage as representing true or false. A variable whose value can be either 1 or 0 is called a Boolean variable. AND, OR, and NOT are the basic Boolean operations. We can express Boolean functions with either an expression or a truth table. Every Boolean expression can be converted to a circuit. Now, we ll look at how Boolean algebra can help simplify expressions, which in turn will lead to simpler circuits. CS231 Boolean Algebra 3

4 Boolean Algebra Last time we talked about Boolean functions, Boolean expressions, and truth tables. Today we ll learn how to how use Boolean algebra to simplify Booleans expressions. Last time, we saw this expression and converted it to a circuit: (x + y )z + x Can we make this circuit better? Cheaper: fewer gates Faster: fewer delays from inputs to outputs June 11, Howard Huang4

5 Expression simplification Normal mathematical expressions can be simplified using the laws of algebra For binary systems, we can use Boolean algebra, which is superficially similar to regular algebra There are many differences, due to having only two values (0 and 1) to work with having a complement operation the OR operation is not the same as addition CS231 Boolean Algebra 5

6 Formal definition of Boolean algebra A Boolean algebra requires A set of elements B, which needs at least two elements (0 and 1) Two binary (two-argument) operations OR and AND A unary (one-argument) operation NOT The axioms below must always be true (textbook, p. 42) The magenta axioms deal with the complement operation Blue axioms (especially 15) are different from regular algebra 1. x + 0 = x 2. x 1 = x 3. x + 1 = 1 4. x 0 = 0 5. x + x = x 6. x x = x 7. x + x = 1 8. x x = 0 9. (x ) = x 10. x + y = y + x 11. xy = yx Commutative 12. x + (y + z) = (x + y) + z 13. x(yz) = (xy)z Associative 14. x(y + z) = xy + xz 15. x + yz = (x + y)(x + z) Distributive 16. (x + y) = x y 17. (xy) = x + y DeMorgan s CS231 Boolean Algebra 6

7 Comments on the axioms The associative laws show that there is no ambiguity about a term such as x + y + z or xyz, so we can introduce multiple-input primitive gates: The left and right columns of axioms are duals exchange all ANDs with ORs, and 0s with 1s The dual of any equation is always true 1. x + 0 = x 2. x 1 = x 3. x + 1 = 1 4. x 0 = 0 5. x + x = x 6. x x = x 7. x + x = 1 8. x x = 0 9. (x ) = x 10. x + y = y + x 11. xy = yx Commutative 12. x + (y + z) = (x + y) + z 13. x(yz) = (xy)z Associative 14. x(y + z) = xy + xz 15. x + yz = (x + y)(x + z) Distributive 16. (x + y) = x y 17. (xy) = x + y DeMorgan s CS231 Boolean Algebra 7

8 Are these axioms for real? We can show that these axioms are valid, given the definitions of AND, OR and NOT x y xy x y x+y x x The first 11 axioms are easy to see from these truth tables alone. For example, x + x = 1 because of the middle two lines below (where y = x ) x y x+y CS231 Boolean Algebra 8

9 Proving the rest of the axioms We can make up truth tables to prove (both parts of) DeMorgan s law For (x + y) = x y, we can make truth tables for (x + y) and for x y x y x + y (x + y) x y x y x y In each table, the columns on the left (x and y) are the inputs. The columns on the right are outputs. In this case, we only care about the columns in blue. The other outputs are just to help us find the blue columns. Since both of the columns in blue are the same, this shows that (x + y) and x y are equivalent CS231 Boolean Algebra 9

10 Simplification with axioms We can now start doing some simplifications x y + xyz + x y = x (y + y) + xyz [ Distributive; x y + x y = x (y + y) ] = x 1 + xyz [ Axiom 7; y + y = 1 ] = x + xyz [ Axiom 2; x 1 = x ] = (x + x)(x + yz) [ Distributive ] = 1 (x + yz) [ Axiom 7; x + x = 1 ] = x + yz [ Axiom 2 ] 1. x + 0 = x 2. x 1 = x 3. x + 1 = 1 4. x 0 = 0 5. x + x = x 6. x x = x 7. x + x = 1 8. x x = 0 9. (x ) = x 10. x + y = y + x 11. xy = yx Commutative 12. x + (y + z) = (x + y) + z 13. x(yz) = (xy)z Associative 14. x(y + z) = xy + xz 15. x + yz = (x + y)(x + z) Distributive 16. (x + y) = x y 17. (xy) = x + y DeMorgan s CS231 Boolean Algebra 10

11 Here are two different but equivalent circuits. In general the one with fewer gates is better : It costs less to build Let s compare the resulting circuits It requires less power But we had to do some work to find the second form CS231 Boolean Algebra 11

12 Some more laws Here are some more useful laws. Notice the duals again! 1. x + xy = x 4. x(x + y) = x 2. xy + xy = x 5. (x + y)(x + y ) = x 3. x + x y = x + y 6. x(x + y) = xy xy + x z + yz = xy + x z (x + y)(x + z)(y + z) = (x + y)(x + z) We can prove these laws by either Making truth tables: Using the axioms: x + x y = (x + x )(x + y) [ Distributive ] = 1 (x + y) [ x + x = 1 ] = x + y [ Axiom 2 ] CS231 Boolean Algebra 12

13 The complement of a function The complement of a function always outputs 0 where the original function outputted 1, and 1 where the original produced 0. In a truth table, we can just exchange 0s and 1s in the output column(s) f(x,y,z) = x(y z + yz) x y z f(x,y,z) x y z f (x,y,z) CS231 Boolean Algebra 13

14 Complementing a function algebraically You can use DeMorgan s law to keep pushing the complements inwards f(x,y,z) = x(y z + yz) f (x,y,z) = ( x(y z + yz) ) [ complement both sides ] = x + (y z + yz) [ because (xy) = x + y ] = x + (y z ) (yz) [ because (x + y) = x y ] = x + (y + z)(y + z ) [ because (xy) = x + y, twice] You can also take the dual of the function, and then complement each literal If f(x,y,z) = x(y z + yz) the dual of f is x + (y + z )(y + z) then complementing each literal gives x + (y + z)(y + z ) so f (x,y,z) = x + (y + z)(y + z ) CS231 Boolean Algebra 14

15 Summary so far So far: A bunch of Boolean algebra trickery for simplifying expressions and circuits The algebra guarantees us that the simplified circuit is equivalent to the original one Next: Introducing some standard forms and terminology An alternative simplification method We ll start using all this stuff to build and analyze bigger, more useful, circuits CS231 Boolean Algebra 15

16 Simplify the expression (x+y )(x+y) + x yz A: x + xy +y x + x yz B: 0 C: x + yz D: x + x yz 1. x + xy = x 4. x(x + y) = x 2. xy + xy = x 5. (x + y)(x + y ) = x 3. x + x y = x + y 6. x(x + y) = xy xy + x z + yz = xy + x z (x + y)(x + z)(y + z) = (x + y)(x + z) 16

17 SIMPLIFY THE EXPRESSION ab c + a bc + b(a c +a (a+c )) A: b+ac B: b+ba+ca C: ca+cb+ba+bc D: abc 1. x + xy = x 4. x(x + y) = x 2. xy + xy = x 5. (x + y)(x + y ) = x 3. x + x y = x + y 6. x(x + y) = xy xy + x z + yz = xy + x z (x + y)(x + z)(y + z) = (x + y)(x + z) CS231 Boolean Algebra 17

Boolean algebra. June 17, Howard Huang 1

Boolean algebra Yesterday we talked about how analog voltages can represent the logical values true and false. We introduced the basic Boolean operations AND, OR and NOT, which can be implemented in hardware

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:

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

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 ):

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

Review: Standard forms of expressions

Karnaugh maps Last time we saw applications of Boolean logic to circuit design. The basic Boolean operations are AND, OR and NOT. These operations can be combined to form complex expressions, which can

Boolean Algebra. P1. The OR operation is closed for all x, y B x + y B

Boolean Algebra A Boolean Algebra is a mathematical system consisting of a set of elements B, two binary operations OR (+) and AND ( ), a unary operation NOT ('), an equality sign (=) to indicate equivalence

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

Spring 2010 CPE231 Digital Logic Section 1 Quiz 1-A. Convert the following numbers from the given base to the other three bases listed in the table:

Section 1 Quiz 1-A Convert the following numbers from the given base to the other three bases listed in the table: Decimal Binary Hexadecimal 1377.140625 10101100001.001001 561.24 454.3125 111000110.0101

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

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

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

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

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

Philadelphia University Faculty of Information Technology Department of Computer Science. Computer Logic Design. By Dareen Hamoudeh.

Philadelphia University Faculty of Information Technology Department of Computer Science Computer Logic Design By Dareen Hamoudeh Dareen Hamoudeh 1 Canonical Forms (Standard Forms of Expression) Minterms

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

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

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)

S1 Teknik Telekomunikasi Fakultas Teknik Elektro FEH2H3 2016/2017

S1 Teknik Telekomunikasi Fakultas Teknik Elektro FEH2H3 2016/2017 Karnaugh Map Karnaugh maps Last time we saw applications of Boolean logic to circuit design. The basic Boolean operations are AND, OR and

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.

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

2.1 Binary Logic and Gates

1 EED2003 Digital Design Presentation 2: Boolean Algebra Asst. Prof.Dr. Ahmet ÖZKURT Asst. Prof.Dr Hakkı T. YALAZAN Based on the Lecture Notes by Jaeyoung Choi choi@comp.ssu.ac.kr Fall 2000 2.1 Binary

Objectives: 1- Bolean Algebra. Eng. Ayman Metwali

Objectives: Chapter 3 : 1- Boolean Algebra Boolean Expressions Boolean Identities Simplification of Boolean Expressions Complements Representing Boolean Functions 2- Logic gates 3- Digital Components 4-

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

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

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

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

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

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.

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

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

Basic circuit analysis and design. Circuit analysis. Write algebraic expressions or make a truth table

Basic circuit analysis and design Circuit analysis Circuit analysis involves figuring out what some circuit does. Every circuit computes some function, which can be described with Boolean expressions or

Permutation Matrices. Permutation Matrices. Permutation Matrices. Permutation Matrices. Isomorphisms of Graphs. 19 Nov 2015

9 Nov 25 A permutation matrix is an n by n matrix with a single in each row and column, elsewhere. If P is a permutation (bijection) on {,2,..,n} let A P be the permutation matrix with A ip(i) =, A ij

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

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

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

BOOLEAN ALGEBRA. 1. State & Verify Laws by using :

BOOLEAN ALGEBRA. State & Verify Laws by using :. State and algebraically verify Absorption Laws. (2) Absorption law states that (i) X + XY = X and (ii) X(X + Y) = X (i) X + XY = X LHS = X + XY = X( + Y)

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

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

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

3. According to universal addressing, what is the address of vertex d? 4. According to universal addressing, what is the address of vertex f?

1. Prove: A full m-ary tree with i internal vertices contains n = mi + 1 vertices. 2. For a full m-ary tree with n vertices, i internal vertices, and l leaves, prove: (i) i = (n 1)/m and l = [(m 1)n +

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

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

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

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

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

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

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

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

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: +

01 Introduction to Digital Logic. ENGR 3410 Computer Architecture Mark L. Chang Fall 2008

Introduction to Digital Logic ENGR 34 Computer Architecture Mark L. Chang Fall 28 Acknowledgements Patterson & Hennessy: Book & Lecture Notes Patterson s 997 course notes (U.C. Berkeley CS 52, 997) Tom

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

DSAS Laboratory no 4. Laboratory 4. Logic forms

Laboratory 4 Logic forms 4.1 Laboratory work goals Going from Boolean functions to Boolean forms. Logic forms equivalence. Boolean forms simplification. Shannon s theorems. Representation in NAND and NOR

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

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

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

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:

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

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 =

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

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

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.

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

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

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

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

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

01 Introduction to Digital Logic. ENGR 3410 Computer Architecture Mark L. Chang Fall 2006

Introduction to Digital Logic ENGR 34 Computer Architecture Mark L. Chang Fall 26 Acknowledgements Patterson & Hennessy: Book & Lecture Notes Patterson s 997 course notes (U.C. Berkeley CS 52, 997) Tom

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?

60-265: Winter ANSWERS Exercise 4 Combinational Circuit Design

60-265: Winter 2010 Computer Architecture I: Digital Design ANSWERS Exercise 4 Combinational Circuit Design Question 1. One-bit Comparator [ 1 mark ] Consider two 1-bit inputs, A and B. If we assume that

Computer Engineering Chapter 3 Boolean Algebra

Computer Engineering Chapter 3 Boolean Algebra Hiroaki Kobayashi 5/30/2011 Ver. 06102011 5/30/2011 Computer Engineering 1 Agenda in Chapter 3 What is Boolean Algebra Basic Boolean/Logical Operations (Operators)

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

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

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

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)

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

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

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

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

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

Section 1.8. Simplifying Expressions

Section 1.8 Simplifying Expressions But, first Commutative property: a + b = b + a; a * b = b * a Associative property: (a + b) + c = a + (b + c) (a * b) * c = a * (b * c) Distributive property: a * (b

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

B.Tech II Year I Semester (R13) Regular Examinations December 2014 DIGITAL LOGIC DESIGN

B.Tech II Year I Semester () Regular Examinations December 2014 (Common to IT and CSE) (a) If 1010 2 + 10 2 = X 10, then X is ----- Write the first 9 decimal digits in base 3. (c) What is meant by don

Chapter 2: Combinational Systems

Uchechukwu Ofoegbu Chapter 2: Combinational Systems Temple University Adapted from Alan Marcovitz s Introduction to Logic and Computer Design Riddle Four switches can be turned on or off. One is the switch

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

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

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

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,

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 4. Combinational Logic. Dr. Abu-Arqoub

Chapter 4 Combinational Logic Introduction N Input Variables Combinational Logic Circuit M Output Variables 2 Design Procedure The problem is stated 2 The number of available input variables & required

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

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)

CS61C : Machine Structures

inst.eecs.berkeley.edu/~cs61c/su06 CS61C : Machine Structures Lecture #14: Combinational Logic, Gates, and State 2006-07-20 CS 61C L14 Combinational Logic (1) Andy Carle What are Machine Structures? Software

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

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

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