Switches: basic element of physical implementations

Size: px
Start display at page:

Download "Switches: basic element of physical implementations"

Transcription

1 Combinational logic Switches Basic logic and truth tables Logic functions Boolean algebra Proofs by re-writing and by perfect induction Winter 200 CSE370 - II - Boolean Algebra Switches: basic element of physical implementations Implementing a simple circuit (arrow shows action if wire changes to ): A Z close switch (if A is or asserted) and turn on light bulb (Z) A Z open switch (if A is 0 or unasserted) and turn off light bulb (Z) Z A Winter 200 CSE370 - II - Boolean Algebra 2

2 Switches (cont d) Compose switches into more complex ones (Boolean functions): AND A B Z A and B A OR Z A or B B Winter 200 CSE370 - II - Boolean Algebra 3 Switching networks Switch settings determine whether or not a conducting path exists to light the light bulb To build larger computations use the light bulb (output of the network) to set other switches (inputs to another network) Winter 200 CSE370 - II - Boolean Algebra 4

3 Transistor networks Modern digital systems are designed in CMOS technology MOS stands for Metal-Oxide on Semiconductor C is for complementary because there are both normally-open and normally-closed switches MOS transistors act as voltage-controlled switches similar, though easier to work with than relays. Winter 200 CSE370 - II - Boolean Algebra 5 MOS transistors MOS transistors have three terminals: drain, gate, and source they act as switches in the following way: if the voltage on the gate terminal is (some amount) higher/lower than the source terminal then a conducting path will be established between the drain and source terminals G G S D S D n-channel open when voltage at G is low closes when: voltage(g) > voltage (S) + ε p-channel closed when voltage at G is low opens when: voltage(g) < voltage (S) ε Winter 200 CSE370 - II - Boolean Algebra 6

4 Most digital logic is CMOS 0V Logic 0.8V Logic 0V.8V.8V 0V 0.3µm.8V.8V Mark Bohr Intel 0V 0V Winter 200 CSE370 - II - Boolean Algebra 7 Multi-input logic gates CMOS logic gates are inverting Easy to implement NAND, NOR, NOT while AND, OR, and Buffer are harder Claude Shannon 938.8V.8V.8V.8V Z Z Z Z 0V 0V Winter 200 CSE370 - II - Boolean Algebra 8

5 Possible logic functions of two variables There are 6 possible functions of 2 input variables: in general, there are 2**(2**n) functions of n inputs F 6 possible functions (F 0 F 5 ) and xor or = nor not ( or ) not not nand not ( and ) Winter 200 CSE370 - II - Boolean Algebra 9 Minimal set of functions Can we implement all logic functions from NOT, NOR, and NAND? For example, implementing and is the same as implementing not ( nand ) In fact, we can do it with only NOR or only NAND NOT is just a NAND or a NOR with both inputs tied together nor nand and NAND and NOR are "duals", that is, its easy to implement one using the other nand not ( (not ) nor (not ) ) nor not ( (not ) nand (not ) ) Winter 200 CSE370 - II - Boolean Algebra 0

6 Boolean algebra An algebraic structure consists of a set of elements B binary operations { +, } and a unary operation { } such that the following axioms hold: George Boole 854. the set B contains at least two elements: a, b 2. closure: a + b is in B a b is in B 3. commutativity: a+b= b+a a b = b a 4. associativity: a + (b + c) = (a + b) + c a (b c) = (a b) c 5. identity: a + 0 = a a = a 6. distributivity: a + (b c) = (a + b) (a + c) a (b + c) = (a b) + (a c) 7. complementarity: a + a = a a = 0 Winter 200 CSE370 - II - Boolean Algebra Logic functions and Boolean algebra Any logic function that can be expressed as a truth table can be written as an expression in Boolean algebra using the operators:, +, and, are Boolean algebra variables ( ) + ( ) ( ) + ( ) = Boolean expression that is true when the variables and have the same value and false, otherwise¾¾ Winter 200 CSE370 - II - Boolean Algebra 2

7 Axioms and theorems of Boolean algebra identity. + 0 = D. = null 2. + = 2D. 0 = 0 idempotency: 3. + = 3D. = involution: 4. ( ) = complementarity: 5. + = 5D. = 0 commutativity: ti it 6. + = + 6D. = associativity: 7. ( + ) + Z = + ( + Z) 7D. ( ) Z = ( Z) distributivity: 8. ( + Z) = ( ) + ( Z) 8D. + ( Z) = ( + ) ( + Z) Winter 200 CSE370 - II - Boolean Algebra 3 Axioms and theorems of Boolean algebra (cont d) uniting: 9. + = 9D. ( + ) ( + ) = absorption: 0. + = 0D. ( + ) =. ( + ) = D. ( ) + = + factoring: 2. ( + ) ( + Z) = 2D. + Z = Z + ( + Z) ( + ) concensus: 3. ( ) + ( Z) + ( Z) = 3D. ( + ) ( + Z) ( + Z) = + Z ( + ) ( +Z) de Morgan s: 4. ( ) =... 4D. (...) = generalized de Morgan s: 5. f (, 2,..., n,0,,+, ) = f(, 2,..., n,,0,,+) Winter 200 CSE370 - II - Boolean Algebra 4

8 Axioms and theorems of Boolean algebra (cont d) Duality a dual of a Boolean expression is derived by replacing by +, + by, 0 by, and by 0, and leaving variables unchanged any theorem that can be proven is thus also proven for its dual! a meta-theorem (a theorem about theorems) duality: generalized duality: 7. f (, 2,..., n,0,,+, ) f(, 2,..., n,,0,,+) Different than demorgan s Law this is a statement about theorems this is not a way to manipulate (re-write) expressions Winter 200 CSE370 - II - Boolean Algebra 5 Proving theorems (rewriting) Using the laws of Boolean algebra: e.g., prove the theorem: + = distributivity (8) complementarity (5) identity (D) + = ( + ) ( + ) = () () = e.g., prove the theorem: + = identity (D) distributivity (8) identity (2) identity (D) + = + + = ( + ) ( + ) = () () = Winter 200 CSE370 - II - Boolean Algebra 6

9 Activity Prove consensus theorem using the laws of Boolean algebra: ( ) + ( Z) + ( Z) = + Z ( ) + ( Z) + ( Z) identity ( ) + () ( Z) + ( Z) complementarity ( ) + ( + ) ( Z) + ( Z) distributivity ( ) + ( Z) + ( Z) + ( Z) commutativity ( ) + ( Z) + ( Z) + ( Z) factoring ( ) ( + Z) + ( Z) ( + ) null ( ) () + ( Z) () identity ( ) + ( Z) identity. + 0 = D. = null 2. + = 2D. 0 = 0 complementarity: 5. + = 5D. = 0 commutativity: 6. + = + 6D. = associativity: 7. ( + ) + Z = + ( + Z) 7D. ( ) Z = ( Z) distributivity: 8. ( + Z) = ( ) + ( Z) 8D. + ( Z) = ( + ) ( + Z) factoring: 2. ( + ) ( + Z) = Z + 2D. + Z = ( + Z) ( + ) Winter 200 CSE370 - II - Boolean Algebra 7 Proving theorems (perfect induction) Using perfect induction (complete truth table): e.g., de Morgan s: ( + ) = NOR is equivalent to AND with inputs complemented ( ) = + NAND is equivalent to OR with inputs complemented ( + ) ( ) Winter 200 CSE370 - II - Boolean Algebra 8

10 A simple example: -bit binary adder Cout Cin Inputs: A, B, Carry-in Outputs: Sum, Carry-out A A A A A B B B B B S S S S S A B Cin Cout S A S B Cout Cin S = A B Cin + A B Cin + A B Cin + A B Cin Cout = A B Cin + A B Cin + A B Cin + A B Cin Winter 200 CSE370 - II Boolean Algebra 9

Combinational logic. Possible logic functions of two variables. Minimal set of functions. Cost of different logic functions.

Combinational logic. Possible logic functions of two variables. Minimal set of functions. Cost of different logic functions. Combinational logic Possible logic functions of two variables Logic functions, truth tables, and switches NOT, ND, OR, NND, NOR, OR,... Minimal set xioms and theorems of oolean algebra Proofs by re-writing

More information

EECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits)

EECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits) EECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits) September 5, 2002 John Wawrzynek Fall 2002 EECS150 Lec4-bool1 Page 1, 9/5 9am Outline Review of

More information

EECS Variable Logic Functions

EECS Variable Logic Functions EECS150 Section 1 Introduction to Combinational Logic Fall 2001 2-Variable Logic Functions There are 16 possible functions of 2 input variables: in general, there are 2**(2**n) functions of n inputs X

More information

Outline. EECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits) Combinational Logic (CL) Defined

Outline. EECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits) Combinational Logic (CL) Defined EECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits) January 30, 2003 John Wawrzynek Outline Review of three representations for combinational logic:

More information

Combinational Logic. Review of Combinational Logic 1

Combinational Logic. Review of Combinational Logic 1 Combinational Logic! Switches -> Boolean algebra! Representation of Boolean functions! Logic circuit elements - logic gates! Regular logic structures! Timing behavior of combinational logic! HDLs and combinational

More information

Intro To Digital Logic

Intro To Digital Logic Intro To Digital Logic 1 Announcements... Project 2.2 out But delayed till after the midterm Midterm in a week Covers up to last lecture + next week's homework & lab Nick goes "H-Bomb of Justice" About

More information

Lecture A: Logic Design and Gates

Lecture A: Logic Design and Gates Lecture A: Logic Design and Gates Syllabus My office hours 9.15-10.35am T,Th or gchoi@ece.tamu.edu 333G WERC Text: Brown and Vranesic Fundamentals of Digital Logic,» Buy it.. Or borrow it» Other book:

More information

Chapter 2: Switching Algebra and Logic Circuits

Chapter 2: Switching Algebra and Logic Circuits Chapter 2: Switching Algebra and Logic Circuits Formal Foundation of Digital Design In 1854 George Boole published An investigation into the Laws of Thoughts Algebraic system with two values 0 and 1 Used

More information

Possible logic functions of two variables

Possible logic functions of two variables ombinational logic asic logic oolean algebra, proofs by re-writing, proofs by perfect induction logic functions, truth tables, and switches NOT, ND, OR, NND, NOR, OR,..., minimal set Logic realization

More information

CS 226: Digital Logic Design

CS 226: Digital Logic Design CS 226: Digital Logic Design 0 1 1 I S 0 1 0 S Department of Computer Science and Engineering, Indian Institute of Technology Bombay. 1 of 29 Objectives In this lecture we will introduce: 1. Logic functions

More information

Chapter 2 Combinational logic

Chapter 2 Combinational logic Chapter 2 Combinational logic Chapter 2 is very easy. I presume you already took discrete mathemtics. The major part of chapter 2 is boolean algebra. II - Combinational Logic Copyright 24, Gaetano Borriello

More information

Ch 2. Combinational Logic. II - Combinational Logic Contemporary Logic Design 1

Ch 2. Combinational Logic. II - Combinational Logic Contemporary Logic Design 1 Ch 2. Combinational Logic II - Combinational Logic Contemporary Logic Design 1 Combinational logic Define The kind of digital system whose output behavior depends only on the current inputs memoryless:

More information

Lab 1 starts this week: go to your session

Lab 1 starts this week: go to your session Lecture 3: Boolean Algebra Logistics Class email sign up Homework 1 due on Wednesday Lab 1 starts this week: go to your session Last lecture --- Numbers Binary numbers Base conversion Number systems for

More information

EECS150 - Digital Design Lecture 19 - Combinational Logic Circuits : A Deep Dive

EECS150 - Digital Design Lecture 19 - Combinational Logic Circuits : A Deep Dive EECS150 - Digital Design Lecture 19 - Combinational Logic Circuits : A Deep Dive March 30, 2010 John Wawrzynek Spring 2010 EECS150 - Lec19-cl1 Page 1 Boolean Algebra I (Representations of Combinational

More information

Combinational Logic Design Principles

Combinational Logic Design Principles Combinational Logic Design Principles Switching algebra Doru Todinca Department of Computers Politehnica University of Timisoara Outline Introduction Switching algebra Axioms of switching algebra Theorems

More information

Digital Circuit And Logic Design I. Lecture 3

Digital Circuit And Logic Design I. Lecture 3 Digital Circuit And Logic Design I Lecture 3 Outline Combinational Logic Design Principles (). Introduction 2. Switching algebra 3. Combinational-circuit analysis 4. Combinational-circuit synthesis Panupong

More information

Boolean Algebra. The Building Blocks of Digital Logic Design. Section. Section Overview. Binary Operations and Their Representation.

Boolean Algebra. The Building Blocks of Digital Logic Design. Section. Section Overview. Binary Operations and Their Representation. Section 3 Boolean Algebra The Building Blocks of Digital Logic Design Section Overview Binary Operations (AND, OR, NOT), Basic laws, Proof by Perfect Induction, De Morgan s Theorem, Canonical and Standard

More information

Computer Organization: Boolean Logic

Computer Organization: Boolean Logic Computer Organization: Boolean Logic Representing and Manipulating Data Last Unit How to represent data as a sequence of bits How to interpret bit representations Use of levels of abstraction in representing

More information

Boolean algebra. Examples of these individual laws of Boolean, rules and theorems for Boolean algebra are given in the following table.

Boolean algebra. Examples of these individual laws of Boolean, rules and theorems for Boolean algebra are given in the following table. The Laws of Boolean Boolean algebra As well as the logic symbols 0 and 1 being used to represent a digital input or output, we can also use them as constants for a permanently Open or Closed circuit or

More information

Digital Logic (2) Boolean Algebra

Digital Logic (2) Boolean Algebra Digital Logic (2) Boolean Algebra Boolean algebra is the mathematics of digital systems. It was developed in 1850 s by George Boole. We will use Boolean algebra to minimize logic expressions. Karnaugh

More information

Floating Point Representation and Digital Logic. Lecture 11 CS301

Floating Point Representation and Digital Logic. Lecture 11 CS301 Floating Point Representation and Digital Logic Lecture 11 CS301 Administrative Daily Review of today s lecture w Due tomorrow (10/4) at 8am Lab #3 due Friday (9/7) 1:29pm HW #5 assigned w Due Monday 10/8

More information

EEE130 Digital Electronics I Lecture #4

EEE130 Digital Electronics I Lecture #4 EEE130 Digital Electronics I Lecture #4 - Boolean Algebra and Logic Simplification - By Dr. Shahrel A. Suandi Topics to be discussed 4-1 Boolean Operations and Expressions 4-2 Laws and Rules of Boolean

More information

4 Switching Algebra 4.1 Axioms; Signals and Switching Algebra

4 Switching Algebra 4.1 Axioms; Signals and Switching Algebra 4 Switching Algebra 4.1 Axioms; Signals and Switching Algebra To design a digital circuit that will perform a required function, it is necessary to manipulate and combine the various input signals in certain

More information

CS61c: Representations of Combinational Logic Circuits

CS61c: Representations of Combinational Logic Circuits CS61c: Representations of Combinational Logic Circuits J. Wawrzynek March 5, 2003 1 Introduction Recall that synchronous systems are composed of two basic types of circuits, combination logic circuits,

More information

ECE 545 Digital System Design with VHDL Lecture 1. Digital Logic Refresher Part A Combinational Logic Building Blocks

ECE 545 Digital System Design with VHDL Lecture 1. Digital Logic Refresher Part A Combinational Logic Building Blocks ECE 545 Digital System Design with VHDL Lecture Digital Logic Refresher Part A Combinational Logic Building Blocks Lecture Roadmap Combinational Logic Basic Logic Review Basic Gates De Morgan s Law Combinational

More information

Chapter 2 Combinational Logic Circuits

Chapter 2 Combinational Logic Circuits Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 1 Gate Circuits and Boolean Equations Chapter 2 - Part 1 2 Chapter 2 - Part 1 3 Chapter 2 - Part 1 4 Chapter 2 - Part

More information

2 Application of Boolean Algebra Theorems (15 Points - graded for completion only)

2 Application of Boolean Algebra Theorems (15 Points - graded for completion only) CSE140 HW1 Solution (100 Points) 1 Introduction The purpose of this assignment is three-fold. First, it aims to help you practice the application of Boolean Algebra theorems to transform and reduce Boolean

More information

CSE140: Components and Design Techniques for Digital Systems. Introduction. Instructor: Mohsen Imani

CSE140: Components and Design Techniques for Digital Systems. Introduction. Instructor: Mohsen Imani CSE4: Components and Design Techniques for Digital Systems Introduction Instructor: Mohsen Imani Slides from: Prof.Tajana Simunic Rosing & Dr.Pietro Mercati Welcome to CSE 4! Instructor: Mohsen Imani Email:

More information

2009 Spring CS211 Digital Systems & Lab CHAPTER 2: INTRODUCTION TO LOGIC CIRCUITS

2009 Spring CS211 Digital Systems & Lab CHAPTER 2: INTRODUCTION TO LOGIC CIRCUITS CHAPTER 2: INTRODUCTION TO LOGIC CIRCUITS What will we learn? 2 Logic functions and circuits Boolean Algebra Logic gates and Synthesis CAD tools and VHDL Read Section 2.9 and 2.0 Terminology 3 Digital

More information

Binary Logic and Gates. Our objective is to learn how to design digital circuits.

Binary Logic and Gates. Our objective is to learn how to design digital circuits. Binary Logic and Gates Introduction Our objective is to learn how to design digital circuits. These circuits use binary systems. Signals in such binary systems may represent only one of 2 possible values

More information

UC Berkeley College of Engineering, EECS Department CS61C: Representations of Combinational Logic Circuits

UC Berkeley College of Engineering, EECS Department CS61C: Representations of Combinational Logic Circuits 2 Wawrzynek, Garcia 2004 c UCB UC Berkeley College of Engineering, EECS Department CS61C: Representations of Combinational Logic Circuits 1 Introduction Original document by J. Wawrzynek (2003-11-15) Revised

More information

Combinational Logic (mostly review!)

Combinational Logic (mostly review!) ombinational Logic (mostly review!)! Logic functions, truth tables, and switches " NOT, N, OR, NN, NOR, OR,... " Minimal set! xioms and theorems of oolean algebra " Proofs by re-writing " Proofs by perfect

More information

CMSC 313 Lecture 16 Postulates & Theorems of Boolean Algebra Semiconductors CMOS Logic Gates

CMSC 313 Lecture 16 Postulates & Theorems of Boolean Algebra Semiconductors CMOS Logic Gates CMSC 33 Lecture 6 Postulates & Theorems of oolean lgebra Semiconductors CMOS Logic Gates UMC, CMSC33, Richard Chang Last Time Overview of second half of this course Logic gates & symbols

More information

Chapter 2: Boolean Algebra and Logic Gates

Chapter 2: Boolean Algebra and Logic Gates Chapter 2: Boolean Algebra and Logic Gates Mathematical methods that simplify binary logics or circuits rely primarily on Boolean algebra. Boolean algebra: a set of elements, a set of operators, and a

More information

Number System. Decimal to binary Binary to Decimal Binary to octal Binary to hexadecimal Hexadecimal to binary Octal to binary

Number System. Decimal to binary Binary to Decimal Binary to octal Binary to hexadecimal Hexadecimal to binary Octal to binary Number System Decimal to binary Binary to Decimal Binary to octal Binary to hexadecimal Hexadecimal to binary Octal to binary BOOLEAN ALGEBRA BOOLEAN LOGIC OPERATIONS Logical AND Logical OR Logical COMPLEMENTATION

More information

CHAPTER 2 BOOLEAN ALGEBRA

CHAPTER 2 BOOLEAN ALGEBRA CHAPTER 2 BOOLEAN ALGEBRA This chapter in the book includes: Objectives Study Guide 2.1 Introduction 2.2 Basic Operations 2.3 Boolean Expressions and Truth Tables 2.4 Basic Theorems 2.5 Commutative, Associative,

More information

BOOLEAN LOGIC. By- Neha Tyagi PGT CS KV 5 Jaipur II Shift, Jaipur Region. Based on CBSE curriculum Class 11. Neha Tyagi, KV 5 Jaipur II Shift

BOOLEAN LOGIC. By- Neha Tyagi PGT CS KV 5 Jaipur II Shift, Jaipur Region. Based on CBSE curriculum Class 11. Neha Tyagi, KV 5 Jaipur II Shift BOOLEAN LOGIC Based on CBSE curriculum Class 11 By- Neha Tyagi PGT CS KV 5 Jaipur II Shift, Jaipur Region Neha Tyagi, KV 5 Jaipur II Shift Introduction Boolean Logic, also known as boolean algebra was

More information

E&CE 223 Digital Circuits & Systems. Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev

E&CE 223 Digital Circuits & Systems. Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev E&CE 223 Digital Circuits & Systems Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev 4 of 92 Section 2: Boolean Algebra & Logic Gates Major topics Boolean algebra NAND & NOR gates Boolean

More information

Discrete Mathematics. CS204: Spring, Jong C. Park Computer Science Department KAIST

Discrete Mathematics. CS204: Spring, Jong C. Park Computer Science Department KAIST Discrete Mathematics CS204: Spring, 2008 Jong C. Park Computer Science Department KAIST Today s Topics Combinatorial Circuits Properties of Combinatorial Circuits Boolean Algebras Boolean Functions and

More information

ELEC Digital Logic Circuits Fall 2014 Boolean Algebra (Chapter 2)

ELEC Digital Logic Circuits Fall 2014 Boolean Algebra (Chapter 2) ELEC 2200-002 Digital Logic Circuits Fall 2014 Boolean Algebra (Chapter 2) Vishwani D. Agrawal James J. Danaher Professor Department of Electrical and Computer Engineering Auburn University, Auburn, AL

More information

Combinational logic. Possible logic functions of two variables. Minimal set of functions. Cost of different logic functions

Combinational logic. Possible logic functions of two variables. Minimal set of functions. Cost of different logic functions ombinational logic Possible logic functions of two variables asic logic oolean algebra, proofs by re-writing, proofs by perfect induction Logic functions, truth tables, and switches NOT, N, OR, NN,, OR,...,

More information

Ex: Boolean expression for majority function F = A'BC + AB'C + ABC ' + ABC.

Ex: Boolean expression for majority function F = A'BC + AB'C + ABC ' + ABC. Boolean Expression Forms: Sum-of-products (SOP) Write an AND term for each input combination that produces a 1 output. Write the input variable if its value is 1; write its complement otherwise. OR the

More information

Logic Design. Chapter 2: Introduction to Logic Circuits

Logic Design. Chapter 2: Introduction to Logic Circuits Logic Design Chapter 2: Introduction to Logic Circuits Introduction Logic circuits perform operation on digital signal Digital signal: signal values are restricted to a few discrete values Binary logic

More information

CS 121 Digital Logic Design. Chapter 2. Teacher Assistant. Hanin Abdulrahman

CS 121 Digital Logic Design. Chapter 2. Teacher Assistant. Hanin Abdulrahman CS 121 Digital Logic Design Chapter 2 Teacher Assistant Hanin Abdulrahman 1 2 Outline 2.2 Basic Definitions 2.3 Axiomatic Definition of Boolean Algebra. 2.4 Basic Theorems and Properties 2.5 Boolean Functions

More information

XOR - XNOR Gates. The graphic symbol and truth table of XOR gate is shown in the figure.

XOR - XNOR Gates. The graphic symbol and truth table of XOR gate is shown in the figure. XOR - XNOR Gates Lesson Objectives: In addition to AND, OR, NOT, NAND and NOR gates, exclusive-or (XOR) and exclusive-nor (XNOR) gates are also used in the design of digital circuits. These have special

More information

Implementation of Boolean Logic by Digital Circuits

Implementation of Boolean Logic by Digital Circuits Implementation of Boolean Logic by Digital Circuits We now consider the use of electronic circuits to implement Boolean functions and arithmetic functions that can be derived from these Boolean functions.

More information

Boolean Algebra & Digital Logic

Boolean Algebra & Digital Logic Boolean Algebra & Digital Logic Boolean algebra was developed by the Englishman George Boole, who published the basic principles in the 1854 treatise An Investigation of the Laws of Thought on Which to

More information

2. Associative Law: A binary operator * on a set S is said to be associated whenever (A*B)*C = A*(B*C) for all A,B,C S.

2. Associative Law: A binary operator * on a set S is said to be associated whenever (A*B)*C = A*(B*C) for all A,B,C S. BOOLEAN ALGEBRA 2.1 Introduction Binary logic deals with variables that have two discrete values: 1 for TRUE and 0 for FALSE. A simple switching circuit containing active elements such as a diode and transistor

More information

Digital Logic. CS211 Computer Architecture. l Topics. l Transistors (Design & Types) l Logic Gates. l Combinational Circuits.

Digital Logic. CS211 Computer Architecture. l Topics. l Transistors (Design & Types) l Logic Gates. l Combinational Circuits. CS211 Computer Architecture Digital Logic l Topics l Transistors (Design & Types) l Logic Gates l Combinational Circuits l K-Maps Figures & Tables borrowed from:! http://www.allaboutcircuits.com/vol_4/index.html!

More information

CMSC 313 Lecture 17 Postulates & Theorems of Boolean Algebra Semiconductors CMOS Logic Gates

CMSC 313 Lecture 17 Postulates & Theorems of Boolean Algebra Semiconductors CMOS Logic Gates CMSC 313 Lecture 17 Postulates & Theorems of Boolean Algebra Semiconductors CMOS Logic Gates UMBC, CMSC313, Richard Chang Last Time Overview of second half of this course Logic gates &

More information

CprE 281: Digital Logic

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

More information

E&CE 223 Digital Circuits & Systems. Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev. Section 2: Boolean Algebra & Logic Gates

E&CE 223 Digital Circuits & Systems. Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev. Section 2: Boolean Algebra & Logic Gates Digital Circuits & Systems Lecture Transparencies (Boolean lgebra & Logic Gates) M. Sachdev 4 of 92 Section 2: Boolean lgebra & Logic Gates Major topics Boolean algebra NND & NOR gates Boolean algebra

More information

BOOLEAN ALGEBRA. Introduction. 1854: Logical algebra was published by George Boole known today as Boolean Algebra

BOOLEAN ALGEBRA. Introduction. 1854: Logical algebra was published by George Boole known today as Boolean Algebra BOOLEAN ALGEBRA Introduction 1854: Logical algebra was published by George Boole known today as Boolean Algebra It s a convenient way and systematic way of expressing and analyzing the operation of logic

More information

L2: Combinational Logic Design (Construction and Boolean Algebra)

L2: Combinational Logic Design (Construction and Boolean Algebra) L2: Combinational Logic Design (Construction and Boolean Algebra) Acknowledgements: Lecture material adapted from Chapter 2 of R. Katz, G. Borriello, Contemporary Logic Design (second edition), Pearson

More information

Every time has a value associated with it, not just some times. A variable can take on any value within a range

Every time has a value associated with it, not just some times. A variable can take on any value within a range Digital Logic Circuits Binary Logic and Gates Logic Simulation Boolean Algebra NAND/NOR and XOR gates Decoder fundamentals Half Adder, Full Adder, Ripple Carry Adder Analog vs Digital Analog Continuous»

More information

Slides for Lecture 10

Slides for Lecture 10 Slides for Lecture 10 ENEL 353: Digital Circuits Fall 2013 Term Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary 30 September, 2013 ENEL 353

More information

3 Boolean Algebra 3.1 BOOLEAN ALGEBRA

3 Boolean Algebra 3.1 BOOLEAN ALGEBRA 3 Boolean Algebra 3.1 BOOLEAN ALGEBRA In 1854, George Boole introduced the following formalism which eventually became Boolean Algebra. Definition. An algebraic system consisting of a set B of elements

More information

Slide Set 3. for ENEL 353 Fall Steve Norman, PhD, PEng. Electrical & Computer Engineering Schulich School of Engineering University of Calgary

Slide Set 3. for ENEL 353 Fall Steve Norman, PhD, PEng. Electrical & Computer Engineering Schulich School of Engineering University of Calgary Slide Set 3 for ENEL 353 Fall 2016 Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary Fall Term, 2016 SN s ENEL 353 Fall 2016 Slide Set 3 slide

More information

WEEK 2.1 BOOLEAN ALGEBRA

WEEK 2.1 BOOLEAN ALGEBRA WEEK 2.1 BOOLEAN ALGEBRA 1 Boolean Algebra Boolean algebra was introduced in 1854 by George Boole and in 1938 was shown by C. E. Shannon to be useful for manipulating Boolean logic functions. The postulates

More information

BOOLEAN ALGEBRA INTRODUCTION SUBSETS

BOOLEAN ALGEBRA INTRODUCTION SUBSETS BOOLEAN ALGEBRA M. Ragheb 1/294/2018 INTRODUCTION Modern algebra is centered around the concept of an algebraic system: A, consisting of a set of elements: ai, i=1, 2,, which are combined by a set of operations

More information

Chapter 2: Princess Sumaya Univ. Computer Engineering Dept.

Chapter 2: Princess Sumaya Univ. Computer Engineering Dept. hapter 2: Princess Sumaya Univ. omputer Engineering Dept. Basic Definitions Binary Operators AND z = x y = x y z=1 if x=1 AND y=1 OR z = x + y z=1 if x=1 OR y=1 NOT z = x = x z=1 if x=0 Boolean Algebra

More information

ECE 545 Digital System Design with VHDL Lecture 1A. Digital Logic Refresher Part A Combinational Logic Building Blocks

ECE 545 Digital System Design with VHDL Lecture 1A. Digital Logic Refresher Part A Combinational Logic Building Blocks ECE 545 Digital System Design with VHDL Lecture A Digital Logic Refresher Part A Combinational Logic Building Blocks Lecture Roadmap Combinational Logic Basic Logic Review Basic Gates De Morgan s Laws

More information

CHAPTER III BOOLEAN ALGEBRA

CHAPTER III BOOLEAN ALGEBRA CHAPTER III- CHAPTER III CHAPTER III R.M. Dansereau; v.. CHAPTER III-2 BOOLEAN VALUES INTRODUCTION BOOLEAN VALUES Boolean algebra is a form of algebra that deals with single digit binary values and variables.

More information

Boolean Algebra & Logic Gates. By : Ali Mustafa

Boolean Algebra & Logic Gates. By : Ali Mustafa Boolean Algebra & Logic Gates By : Ali Mustafa Digital Logic Gates There are three fundamental logical operations, from which all other functions, no matter how complex, can be derived. These Basic functions

More information

Logic Gates and Boolean Algebra

Logic Gates and Boolean Algebra Logic Gates and oolean lgebra The ridge etween Symbolic Logic nd Electronic Digital Computing Compiled y: Muzammil hmad Khan mukhan@ssuet.edu.pk asic Logic Functions and or nand nor xor xnor not 2 Logic

More information

CMPE12 - Notes chapter 1. Digital Logic. (Textbook Chapter 3)

CMPE12 - Notes chapter 1. Digital Logic. (Textbook Chapter 3) CMPE12 - Notes chapter 1 Digital Logic (Textbook Chapter 3) Transistor: Building Block of Computers Microprocessors contain TONS of transistors Intel Montecito (2005): 1.72 billion Intel Pentium 4 (2000):

More information

Week-I. Combinational Logic & Circuits

Week-I. Combinational Logic & Circuits Week-I Combinational Logic & Circuits Overview Binary logic operations and gates Switching algebra Algebraic Minimization Standard forms Karnaugh Map Minimization Other logic operators IC families and

More information

. T SHREE MAHAPRABHU PUBLIC SCHOOL & COLLEGE NOTES FOR BOARD EXAMINATION SUBJECT COMPUTER SCIENCE (Code: 083) Boolean Algebra

. T SHREE MAHAPRABHU PUBLIC SCHOOL & COLLEGE NOTES FOR BOARD EXAMINATION SUBJECT COMPUTER SCIENCE (Code: 083) Boolean Algebra . T SHREE MAHAPRABHU PUBLIC SCHOOL & COLLEGE NOTES FOR BOARD EXAMINATION 2016-17 SUBJECT COMPUTER SCIENCE (Code: 083) Boolean Algebra Introduction to Boolean Algebra Boolean algebra which deals with two-valued

More information

Unit 8A Computer Organization. Boolean Logic and Gates

Unit 8A Computer Organization. Boolean Logic and Gates Unit 8A Computer Organization Boolean Logic and Gates Announcements Bring ear buds or headphones to lab! 15110 Principles of Computing, Carnegie Mellon University - CORTINA 2 Representing and Manipulating

More information

Chapter 2 Boolean Algebra and Logic Gates

Chapter 2 Boolean Algebra and Logic Gates Chapter 2 Boolean Algebra and Logic Gates The most common postulates used to formulate various algebraic structures are: 1. Closure. N={1,2,3,4 }, for any a,b N we obtain a unique c N by the operation

More information

Binary Logic and Gates

Binary Logic and Gates 1 COE 202- Digital Logic Binary Logic and Gates Dr. Abdulaziz Y. Barnawi COE Department KFUPM 2 Outline Introduction Boolean Algebra Elements of Boolean Algebra (Binary Logic) Logic Operations & Logic

More information

COSC 243. Introduction to Logic And Combinatorial Logic. Lecture 4 - Introduction to Logic and Combinatorial Logic. COSC 243 (Computer Architecture)

COSC 243. Introduction to Logic And Combinatorial Logic. Lecture 4 - Introduction to Logic and Combinatorial Logic. COSC 243 (Computer Architecture) COSC 243 Introduction to Logic And Combinatorial Logic 1 Overview This Lecture Introduction to Digital Logic Gates Boolean algebra Combinatorial Logic Source: Chapter 11 (10 th edition) Source: J.R. Gregg,

More information

CHAPTER III BOOLEAN ALGEBRA

CHAPTER III BOOLEAN ALGEBRA CHAPTER III- CHAPTER III CHAPTER III R.M. Dansereau; v.. CHAPTER III-2 BOOLEAN VALUES INTRODUCTION BOOLEAN VALUES Boolean algebra is a form of algebra that deals with single digit binary values and variables.

More information

Boolean Algebra and logic gates

Boolean Algebra and logic gates Boolean Algebra and logic gates Luis Entrena, Celia López, Mario García, Enrique San Millán Universidad Carlos III de Madrid 1 Outline l Postulates and fundamental properties of Boolean Algebra l Boolean

More information

DIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA

DIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA DIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA 1 Learning Objectives Understand the basic operations and laws of Boolean algebra. Relate these operations and laws to circuits composed of AND gates, OR gates, INVERTERS

More information

Lecture 2 Review on Digital Logic (Part 1)

Lecture 2 Review on Digital Logic (Part 1) Lecture 2 Review on Digital Logic (Part 1) Xuan Silvia Zhang Washington University in St. Louis http://classes.engineering.wustl.edu/ese461/ Grading Engagement 5% Review Quiz 10% Homework 10% Labs 40%

More information

Boolean Algebra. Philipp Koehn. 9 September 2016

Boolean Algebra. Philipp Koehn. 9 September 2016 Boolean Algebra Philipp Koehn 9 September 2016 Core Boolean Operators 1 AND OR NOT A B A and B 0 0 0 0 1 0 1 0 0 1 1 1 A B A or B 0 0 0 0 1 1 1 0 1 1 1 1 A not A 0 1 1 0 AND OR NOT 2 Boolean algebra Boolean

More information

Unit 2 Boolean Algebra

Unit 2 Boolean Algebra Unit 2 Boolean Algebra 1. Developed by George Boole in 1847 2. Applied to the Design of Switching Circuit by Claude Shannon in 1939 Department of Communication Engineering, NCTU 1 2.1 Basic Operations

More information

CS/COE0447: Computer Organization and Assembly Language

CS/COE0447: Computer Organization and Assembly Language CS/COE0447: Computer Organization and Assembly Language Logic Design Introduction (Brief?) Appendix B: The Basics of Logic Design Dept. of Computer Science Logic design? Digital hardware is implemented

More information

CS 61C: Great Ideas in Computer Architecture Synchronous Digital Systems

CS 61C: Great Ideas in Computer Architecture Synchronous Digital Systems CS 61C: Great Ideas in Computer Architecture Synchronous Digital Systems Instructors: Krste Asanovic & Vladimir Stojanovic h>p://inst.eecs.berkeley.edu/~cs61c/sp15 1 So3ware Parallel Requests Assigned

More information

Logic and Boolean algebra

Logic and Boolean algebra Computer Mathematics Week 7 Logic and Boolean algebra College of Information Science and Engineering Ritsumeikan University last week coding theory channel coding information theory concept Hamming distance

More information

Administrative Notes. Chapter 2 <9>

Administrative Notes. Chapter 2 <9> Administrative Notes Note: New homework instructions starting with HW03 Homework is due at the beginning of class Homework must be organized, legible (messy is not), and stapled to be graded Chapter 2

More information

CS/COE1541: Introduction to Computer Architecture. Logic Design Review. Sangyeun Cho. Computer Science Department University of Pittsburgh

CS/COE1541: Introduction to Computer Architecture. Logic Design Review. Sangyeun Cho. Computer Science Department University of Pittsburgh CS/COE54: Introduction to Computer Architecture Logic Design Review Sangyeun Cho Computer Science Department Logic design? Digital hardware is implemented by way of logic design Digital circuits process

More information

Lecture 3: Boolean Algebra

Lecture 3: Boolean Algebra Lecture 3: Boolean Algebra 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 Overview

More information

Unit 2 Boolean Algebra

Unit 2 Boolean Algebra Unit 2 Boolean Algebra 2.1 Introduction We will use variables like x or y to represent inputs and outputs (I/O) of a switching circuit. Since most switching circuits are 2 state devices (having only 2

More information

Number System conversions

Number System conversions Number System conversions Number Systems The system used to count discrete units is called number system. There are four systems of arithmetic which are often used in digital electronics. Decimal Number

More information

Chapter 2 Boolean Algebra and Logic Gates

Chapter 2 Boolean Algebra and Logic Gates Chapter 2 Boolean Algebra and Logic Gates Huntington Postulates 1. (a) Closure w.r.t. +. (b) Closure w.r.t.. 2. (a) Identity element 0 w.r.t. +. x + 0 = 0 + x = x. (b) Identity element 1 w.r.t.. x 1 =

More information

ENG2410 Digital Design Combinational Logic Circuits

ENG2410 Digital Design Combinational Logic Circuits ENG240 Digital Design Combinational Logic Circuits Fall 207 S. Areibi School of Engineering University of Guelph Binary variables Binary Logic Can be 0 or (T or F, low or high) Variables named with single

More information

Chapter 2 Combinational Logic Circuits

Chapter 2 Combinational Logic Circuits Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 3 Additional Gates and Circuits Overview Part 1 Gate Circuits and Boolean Equations Binary Logic and Gates Boolean Algebra

More information

THE LOGIC OF COMPOUND STATEMENTS

THE LOGIC OF COMPOUND STATEMENTS CHAPTER 2 THE LOGIC OF COMPOUND STATEMENTS Copyright Cengage Learning. All rights reserved. SECTION 2.4 Application: Digital Logic Circuits Copyright Cengage Learning. All rights reserved. Application:

More information

E40M. Binary Numbers. M. Horowitz, J. Plummer, R. Howe 1

E40M. Binary Numbers. M. Horowitz, J. Plummer, R. Howe 1 E40M Binary Numbers M. Horowitz, J. Plummer, R. Howe 1 Reading Chapter 5 in the reader A&L 5.6 M. Horowitz, J. Plummer, R. Howe 2 Useless Box Lab Project #2 Adding a computer to the Useless Box alows us

More information

Combinational Logic Fundamentals

Combinational Logic Fundamentals Topic 3: Combinational Logic Fundamentals In this note we will study combinational logic, which is the part of digital logic that uses Boolean algebra. All the concepts presented in combinational logic

More information

KP/Worksheets: Propositional Logic, Boolean Algebra and Computer Hardware Page 1 of 8

KP/Worksheets: Propositional Logic, Boolean Algebra and Computer Hardware Page 1 of 8 KP/Worksheets: Propositional Logic, Boolean Algebra and Computer Hardware Page 1 of 8 Q1. What is a Proposition? Q2. What are Simple and Compound Propositions? Q3. What is a Connective? Q4. What are Sentential

More information

Circuits & Boolean algebra.

Circuits & Boolean algebra. Circuits & Boolean algebra http://xkcd.com/730/ CSCI 255: Introduction to Embedded Systems Keith Vertanen Copyright 2011 Digital circuits Overview How a switch works Building basic gates from switches

More information

CSE 311: Foundations of Computing. Lecture 3: Digital Circuits & Equivalence

CSE 311: Foundations of Computing. Lecture 3: Digital Circuits & Equivalence CSE 311: Foundations of Computing Lecture 3: Digital Circuits & Equivalence Homework #1 You should have received An e-mail from [cse311a/cse311b] with information pointing you to look at Canvas to submit

More information

CHAPTER1: Digital Logic Circuits Combination Circuits

CHAPTER1: Digital Logic Circuits Combination Circuits CS224: Computer Organization S.KHABET CHAPTER1: Digital Logic Circuits Combination Circuits 1 PRIMITIVE LOGIC GATES Each of our basic operations can be implemented in hardware using a primitive logic gate.

More information

Boolean Algebra and Logic Gates

Boolean Algebra and Logic Gates Boolean Algebra and Logic Gates ( 范倫達 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C. Fall, 2017 ldvan@cs.nctu.edu.tw http://www.cs.nctu.edu.tw/~ldvan/ Outlines Basic

More information

Appendix A: Digital Logic. Principles of Computer Architecture. Principles of Computer Architecture by M. Murdocca and V. Heuring

Appendix A: Digital Logic. Principles of Computer Architecture. Principles of Computer Architecture by M. Murdocca and V. Heuring - Principles of Computer rchitecture Miles Murdocca and Vincent Heuring 999 M. Murdocca and V. Heuring -2 Chapter Contents. Introduction.2 Combinational Logic.3 Truth Tables.4 Logic Gates.5 Properties

More information

CMPE12 - Notes chapter 2. Digital Logic. (Textbook Chapters and 2.1)"

CMPE12 - Notes chapter 2. Digital Logic. (Textbook Chapters and 2.1) CMPE12 - Notes chapter 2 Digital Logic (Textbook Chapters 3.1-3.5 and 2.1)" Truth table" The most basic representation of a logic function." Brute force representation listing the output for all possible

More information

control in out in out Figure 1. Binary switch: (a) opened or off; (b) closed or on.

control in out in out Figure 1. Binary switch: (a) opened or off; (b) closed or on. Chapter 2 Digital Circuits Page 1 of 18 2. Digital Circuits Our world is an analog world. Measurements that we make of the physical objects around us are never in discrete units but rather in a continuous

More information