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

Size: px
Start display at page:

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

Transcription

1 CHAPTER 2: INTRODUCTION TO LOGIC CIRCUITS

2 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

3 Terminology 3 Digital it circuits it Signal values restricted to a few discrete values Binary logic circuit two values, 0 and Why do we use the binary circuit? Multi-valued logic More than two values Some flash devices use multi-valued data storage Multi-level cell(mlc) capable of storing 2 bits of data per cell Not covered in this class Analog circuits

4 Example of Binary logic 4 Binary switch open closed Input variable x = 0 x = (a) Two states of a switch S (b) Symbol for a switch x

5 Application of a switch 5 S Battery x Light Output (a) Simple connection to a battery State of the light : L = if the light is on, L=0 (off) The state of the light can be described as a function of the input variable x L(x) = x : logic function

6 Basic two functions 6 Power supply S S L(, ) = or ( x ) Light where L= if =and= =, L=0 otherwise (a) The logical AND function (series connection) S Power supply S Light L(x, ) = + where L=0 if x = x2=0, L= otherwise (b) The logical OR function (parallel connection)

7 Series-Parallel Connection 7 S X S Power supply S X 3 Light X 2 Logic function?

8 Inversion 8 R Power supply x S Light Inverse of a value Complement of a value The NOT operation Several commonly used notations x = x =!x = NOT x = ~x The complement operation can be applied to more complex operations f(, ) = +

9 Truth Tables: AND and OR 9 Useful aid for depicting information involving logic functions Grow exponentially in size with the number of variables

10 Three input AND and OR 0 Table size : n inputs? = 2 n

11 Possible logic functions of two variables 6 possible functions of 2 input variables N inputs : 2**(2**n) X Y F X Y X and Y X Y X xor Y X or Y X nor Y X = Y not X not Y X nand Y Complete Set : a set of function is complete iff every Boolean function can be generated by a combination of the functions {NAND}, {NOR}, {AND, NOT} {AND, OR}?

12 Basic Gates 2 x x 2 x n 2 x n (a) AND gates x x x n x x x n (b) OR gates (c) NOT gate

13 An AND-OR Function 3 S X S Power supply S X 3 Light X 2 f = ( x + x ) x x f = ( x + x ) x 3 2 3

14 Analysis of a Logic Network 4 Analysis and Synthesis x A 0 f B (a) Network that implements f = + x x x f ( x, ) A B (b) Truth table

15 Timing Diagram 5 Timing i Diagram Changes in signals at various points are presented in graphical form Also useful method for indicating the functional behavior x x A B 0 f A 0 B 0 f 0 Time (c) Timing diagram waveform

16 Functionally Equivalent Networks 6 + = + x A 0 f x 0 2 B (a) Network that implements f = x g (d) Network that implements g = x + 2

17 Functionally Equivalent Networks(cont d) 7 A logic function can be implemented with a variety of different networks, probably having different costs This raises an important question How do we find the best implementation for a given function? What is the best implementation? Many techniques for synthesizing logic functions Discussed in Chapter 4 Through the Truth Table Algebraic(Boolean) manipulation of logic expressions One more question Why we call them functionally equivalent?

18 What will we learn? 8 Logic functions and circuits Boolean Algebra Logic gates and Synthesis CAD tools and VHDL Read Section 2.9 and 2.0

19 Boolean Algebra 9 In 849 George Boole A scheme for the algebraic description of processes involved in logical thought and reasoning In the late 930 Claude Shannon showed Boolean algebra provides an effective means of describing circuits built with switches A Boolean algebraic structure consists of A set of element B = {0, } Binary operation { +(OR), (AND) } And a unary operation { (NOT)} Such that the following axioms hold :

20 Axioms and theorems of Boolean Algebra 20 Axioms Single-variable theorems Two-and Three-variable properties p Duality A dual of a Boolean expression is derived by replacing by +, + by, 0 by, and by 0, and leaving variables unchanged Dual of any true statement is also true X + Y + X Y DeMorgan s theorem X Y = X + Y X + Y = X Y Boolean algebra read section 2.5

21 Proof of DeMorgan s Theorem 2

22 Notation and Precedence of Operations 22 Logical sum and logical product operations The OR and AND operations Say simply sum and product Product terms and Sum terms x y : product term (xy) x+y : sum term Precedence In the order: NOT, AND, and then OR

23 From Boolean expressions to logic gates 23 notation ti gate truth th table NOT X X ~X AND X Y XY X^Y OR X+Y X Y NAND X Y NOR X+Y XOR X Y (XY +X Y) XNOR X = Y (XY+X Y )

24 What will we learn? 24 Logic functions and circuits Boolean Algebra Logic gates and Synthesis CAD tools and VHDL Read Section 2.9 and 2.0

25 Logic functions and Boolean Algebra 25 Any logic function that can be expressed as a truth table written as an expression in Boolean algebra Implementing an arbitrary logic function Derive the truth table Create the product term for which h output has a value of Logical sum of these product terms

26 Example 26 f = + + Logic function A function to be synthesized. f (a) Canonical sum-of-products (b) Minimal-cost realization f

27 What is the best realization? 27 Reduce number of inputs Literal : input variables (complemented or not) Can approximate cost of logic gate as 2 transistors per literal Fewer literals means less transistors Fewer inputs implies faster gates Fan-ins (# of gate inputs) are limited in some technologies Rd Reduce number of gates Fewer gates means smaller circuits Reduce number of levels of gates Propagation delays

28 Tradeoffs between circuit delay and size? 28 How do we explore tradeoffs between circuit it delay and size? Tools to generate different solutions Logic minimization: reduce number of gates and complexity Logic optimization: reduction of complexity while trading off against delay (or power consumption) Are all realizations equivalent? Static and dynamic behavior Number of gate levels and structure Delays are different Glitches (hazards) may arise Cost

29 Implementing Boolean functions (Synthesis) 29 Technology independent Canonical forms Two-level forms Multi-level forms Technology choices (or Mapping) Packages of a few gates Two-level programmable logic Multi-level programmable logic

30 Canonical forms 30 Truth table A unique signature of a Boolean function Many alternative gate realizations Many alternative Boolean expressions drawback? Canonical forms Standard forms for a Boolean expression A unique algebraic signature Using Truth table

31 Canonical Forms (cont d) 3 Minterm ANDed product of literals Product term in which each of input variables or its complement appears once Maxterm ORed sum of literals Sum term in which each of input variables or its complement appears once

32 Sum-of-Products Form 32 A function can be represented by Sum of Product form Product of Sum form Disjunctive normal form (minterm expansion) f = m 0 + m +m 2 0 +m 3 = m 0 + m +m 3 = Σm(0,,3) = + + Sum-of-Products (SOP) form Canonical sum-of-products If each product term is a minterm

33 Canonical SOP example 33 f = + + Logic function x2 A function to be synthesized. f (a) Canonical sum-of-products

34 Example 34 Canonical SOP (simplest form??) f(,,x 3 ) = m +m 4 +m 5 +m 6 = x 3 + x 3 SOP x 3 f (a) A minimal sum-of-products realization

35 Product-of-Sums of (POS) Form 35 Conjunctive normal form (Maxterm expansion) f(, ) = m 2 = f(, ) = f = = + = M 2 In the previous example f(,,x 3 ) = M 0 M 2 M 3 M 7 =( +x 3 )( +x 3 ) x 3 f (b) A minimal product-of-sums realization

36 NAND and NOR Logic networks 36 NAND and NOR functions are attractive x Why? 2 x x x 2 x 2 x n 2 x n (a) NAND gates x x x n x n (b) NOR gates

37 DeMorgan s theorem in terms of logic gates 37 Its logic gate interpretation (a) = + (b) + =

38 NAND-NAND Network 38 x 3 x 4 x 3 x 4 x 5 x 5 x 3 x 4 x 5

39 NOR-NOR Network 39 x 3 x 4 x 5 x 3 x 4 x 5 x 3 x 4 x 5

40 Design Example : Three-way light controller 40 Assume that t a large room has three doors and that t a switch near the door controls a light in the room Turn the light on or off by changing the state of any one Design Turn this word statement into a formal specification using a truth table Let,, 2,, and x 3 be the input variables that denote the state of each switch In which case the light is on? The light is off if all switches are open The light is on if any one of the switches is closed The light is off if two (or no) switches are closed The light is on if all switches are closed The required functional behavior can be specified in the truth table

41 Design Example : Three-way light controller 4 f x 3 (a) Sum-of-products realization x 3 f (b) Product-of-sums realization

42 Design Example: Multiplexer Circuit 42 Multiplexer l A circuit that generates an output that exactly reflects the state of one of a number of data inputs, based on the value of one or more selection control inputs Suppose Two sources of data(, )and one output(f) A selection control signal s Assume If s=0, f= If s=, f= These requirements can be specified in the form of a truth table in the next slide

43 Design Example: Multiplexer Circuit 43 s f (s,, ) s s f 0 f (b) Circuit (c) Graphical symbol 0 0 s 0 f (s,, ) (a) Truth table Can we implement it using NAND gate? d) More compact truth-table representation

44 What will we learn? 44 Logic functions and circuits Boolean Algebra Logic gates and Synthesis CAD tools and VHDL Read Section 2.9 and 2.0

45 CAD Tools and VHDL 45 TA will teach you these topics next week Experiment on several circuits using CAD tools and VHDL

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: 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 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

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

EC-121 Digital Logic Design

EC-121 Digital Logic Design EC-121 Digital Logic Design Lecture 2 [Updated on 02-04-18] Boolean Algebra and Logic Gates Dr Hashim Ali Spring 2018 Department of Computer Science and Engineering HITEC University Taxila!1 Overview What

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

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

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

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

Chapter 2 Boolean Algebra and Logic Gates

Chapter 2 Boolean Algebra and Logic Gates Ch1: Digital Systems and Binary Numbers Ch2: Ch3: Gate-Level Minimization Ch4: Combinational Logic Ch5: Synchronous Sequential Logic Ch6: Registers and Counters Switching Theory & Logic Design Prof. Adnan

More information

II. COMBINATIONAL LOGIC DESIGN. - algebra defined on a set of 2 elements, {0, 1}, with binary operators multiply (AND), add (OR), and invert (NOT):

II. COMBINATIONAL LOGIC DESIGN. - algebra defined on a set of 2 elements, {0, 1}, with binary operators multiply (AND), add (OR), and invert (NOT): ENGI 386 Digital Logic II. COMBINATIONAL LOGIC DESIGN Combinational Logic output of digital system is only dependent on current inputs (i.e., no memory) (a) Boolean Algebra - developed by George Boole

More information

Contents. Chapter 2 Digital Circuits Page 1 of 30

Contents. Chapter 2 Digital Circuits Page 1 of 30 Chapter 2 Digital Circuits Page 1 of 30 Contents Contents... 1 2 Digital Circuits... 2 2.1 Binary Numbers... 2 2.2 Binary Switch... 4 2.3 Basic Logic Operators and Logic Expressions... 5 2.4 Truth Tables...

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

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

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

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

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

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

This form sometimes used in logic circuit, example:

This form sometimes used in logic circuit, example: Objectives: 1. Deriving of logical expression form truth tables. 2. Logical expression simplification methods: a. Algebraic manipulation. b. Karnaugh map (k-map). 1. Deriving of logical expression from

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

MC9211 Computer Organization

MC9211 Computer Organization MC92 Computer Organization Unit : Digital Fundamentals Lesson2 : Boolean Algebra and Simplification (KSB) (MCA) (29-2/ODD) (29 - / A&B) Coverage Lesson2 Introduces the basic postulates of Boolean Algebra

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 Charles Kime & Thomas Kaminski 2008 Pearson Education, Inc. (Hyperlinks are active

More information

Switches: basic element of physical implementations

Switches: basic element of physical implementations 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

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

Logic Gate Level. Part 2

Logic Gate Level. Part 2 Logic Gate Level Part 2 Constructing Boolean expression from First method: write nonparenthesized OR of ANDs Each AND is a 1 in the result column of the truth table Works best for table with relatively

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

211: Computer Architecture Summer 2016

211: Computer Architecture Summer 2016 211: Computer Architecture Summer 2016 Liu Liu Topic: Storage Project3 Digital Logic - Storage: Recap - Review: cache hit rate - Project3 - Digital Logic: - truth table => SOP - simplification: Boolean

More information

Combinatorial Logic Design Principles

Combinatorial Logic Design Principles Combinatorial Logic Design Principles ECGR2181 Chapter 4 Notes Logic System Design I 4-1 Boolean algebra a.k.a. switching algebra deals with boolean values -- 0, 1 Positive-logic convention analog voltages

More information

Standard Expression Forms

Standard Expression Forms ThisLecture will cover the following points: Canonical and Standard Forms MinTerms and MaxTerms Digital Logic Families 24 March 2010 Standard Expression Forms Two standard (canonical) expression forms

More information

CPE100: Digital Logic Design I

CPE100: Digital Logic Design I Chapter 2 Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu http://www.ee.unlv.edu/~b1morris/cpe100/ CPE100: Digital Logic Design I Section 1004: Dr. Morris Combinational Logic Design Chapter

More information

Chapter 2. Boolean Algebra and Logic Gates

Chapter 2. Boolean Algebra and Logic Gates Chapter 2 Boolean Algebra and Logic Gates Basic Definitions A binary operator defined on a set S of elements is a rule that assigns, to each pair of elements from S, a unique element from S. The most common

More information

Chapter 2 Boolean Algebra and Logic Gates

Chapter 2 Boolean Algebra and Logic Gates CSA051 - Digital Systems 數位系統導論 Chapter 2 Boolean Algebra and Logic Gates 吳俊興國立高雄大學資訊工程學系 Chapter 2. Boolean Algebra and Logic Gates 2-1 Basic Definitions 2-2 Axiomatic Definition of Boolean Algebra 2-3

More information

ELCT201: DIGITAL LOGIC DESIGN

ELCT201: DIGITAL LOGIC DESIGN ELCT2: DIGITAL LOGIC DESIGN Dr. Eng. Haitham Omran, haitham.omran@guc.edu.eg Dr. Eng. Wassim Alexan, wassim.joseph@guc.edu.eg Lecture 2 Following the slides of Dr. Ahmed H. Madian ذو الحجة 438 ه Winter

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

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

Chapter 4 Optimized Implementation of Logic Functions

Chapter 4 Optimized Implementation of Logic Functions Chapter 4 Optimized Implementation of Logic Functions Logic Minimization Karnaugh Maps Systematic Approach for Logic Minimization Minimization of Incompletely Specified Functions Tabular Method for Minimization

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 Charles Kime & Thomas Kaminski 2008 Pearson Education, Inc. Overview Part 1 Gate

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

EEA051 - Digital Logic 數位邏輯 吳俊興高雄大學資訊工程學系. September 2004

EEA051 - Digital Logic 數位邏輯 吳俊興高雄大學資訊工程學系. September 2004 EEA051 - Digital Logic 數位邏輯 吳俊興高雄大學資訊工程學系 September 2004 Boolean Algebra (formulated by E.V. Huntington, 1904) A set of elements B={0,1} and two binary operators + and Huntington postulates 1. Closure

More information

DIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA (CONT.)

DIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA (CONT.) DIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA (CONT.) 1 Learning Objectives 1. Apply the laws and theorems of Boolean algebra to to the manipulation of algebraic expressions to simplifying an expression, finding

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

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

CSE20: Discrete Mathematics for Computer Science. Lecture Unit 2: Boolan Functions, Logic Circuits, and Implication

CSE20: Discrete Mathematics for Computer Science. Lecture Unit 2: Boolan Functions, Logic Circuits, and Implication CSE20: Discrete Mathematics for Computer Science Lecture Unit 2: Boolan Functions, Logic Circuits, and Implication Disjunctive normal form Example: Let f (x, y, z) =xy z. Write this function in DNF. Minterm

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

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

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

CprE 281: Digital Logic

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

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

ECE 20B, Winter 2003 Introduction to Electrical Engineering, II LECTURE NOTES #2

ECE 20B, Winter 2003 Introduction to Electrical Engineering, II LECTURE NOTES #2 ECE 20B, Winter 2003 Introduction to Electrical Engineering, II LECTURE NOTES #2 Instructor: Andrew B. Kahng (lecture) Email: abk@ucsd.edu Telephone: 858-822-4884 office, 858-353-0550 cell Office: 3802

More information

CprE 281: Digital Logic

CprE 281: Digital Logic CprE 281: Digital Logic Instructor: Alexander Stoytchev http://www.ece.iastate.edu/~alexs/classes/ NAND and NOR Logic Networks CprE 281: Digital Logic Iowa State University, Ames, IA Copyright Alexander

More information

CprE 281: Digital Logic

CprE 281: Digital Logic CprE 281: Digital Logic Instructor: Alexander Stoytchev http://www.ece.iastate.edu/~alexs/classes/ NAND and NOR Logic Networks CprE 281: Digital Logic Iowa State University, Ames, IA Copyright Alexander

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

Chapter 2 (Lect 2) Canonical and Standard Forms. Standard Form. Other Logic Operators Logic Gates. Sum of Minterms Product of Maxterms

Chapter 2 (Lect 2) Canonical and Standard Forms. Standard Form. Other Logic Operators Logic Gates. Sum of Minterms Product of Maxterms Chapter 2 (Lect 2) Canonical and Standard Forms Sum of Minterms Product of Maxterms Standard Form Sum of products Product of sums Other Logic Operators Logic Gates Basic and Multiple Inputs Positive and

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

Functions. Computers take inputs and produce outputs, just like functions in math! Mathematical functions can be expressed in two ways:

Functions. Computers take inputs and produce outputs, just like functions in math! Mathematical functions can be expressed in two ways: Boolean Algebra (1) Functions Computers take inputs and produce outputs, just like functions in math! Mathematical functions can be expressed in two ways: An expression is finite but not unique f(x,y)

More information

Chap 2. Combinational Logic Circuits

Chap 2. Combinational Logic Circuits Overview 2 Chap 2. Combinational Logic Circuits Spring 24 Part Gate Circuits and Boolean Equations Binary Logic and Gates Boolean Algebra Standard Forms Part 2 Circuit Optimization Two-Level Optimization

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

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

ECEN 248: INTRODUCTION TO DIGITAL SYSTEMS DESIGN. Week 2 Dr. Srinivas Shakkottai Dept. of Electrical and Computer Engineering

ECEN 248: INTRODUCTION TO DIGITAL SYSTEMS DESIGN. Week 2 Dr. Srinivas Shakkottai Dept. of Electrical and Computer Engineering ECEN 248: INTRODUCTION TO DIGITAL SYSTEMS DESIGN Week 2 Dr. Srinivas Shakkottai Dept. of Electrical and Computer Engineering Boolean Algebra Boolean Algebra A Boolean algebra is defined with: A set of

More information

XI STANDARD [ COMPUTER SCIENCE ] 5 MARKS STUDY MATERIAL.

XI STANDARD [ COMPUTER SCIENCE ] 5 MARKS STUDY MATERIAL. 2017-18 XI STANDARD [ COMPUTER SCIENCE ] 5 MARKS STUDY MATERIAL HALF ADDER 1. The circuit that performs addition within the Arithmetic and Logic Unit of the CPU are called adders. 2. A unit that adds two

More information

Boolean Algebra and Logic Simplification

Boolean Algebra and Logic Simplification S302 Digital Logic Design Boolean Algebra and Logic Simplification Boolean Analysis of Logic ircuits, evaluating of Boolean expressions, representing the operation of Logic circuits and Boolean expressions

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

Digital Logic. Lecture 5 - Chapter 2. Outline. Other Logic Gates and their uses. Other Logic Operations. CS 2420 Husain Gholoom - lecturer Page 1

Digital Logic. Lecture 5 - Chapter 2. Outline. Other Logic Gates and their uses. Other Logic Operations. CS 2420 Husain Gholoom - lecturer Page 1 Lecture 5 - Chapter 2 Outline Other Logic Gates and their uses Other Logic Operations CS 2420 Husain Gholoom - lecturer Page 1 Digital logic gates CS 2420 Husain Gholoom - lecturer Page 2 Buffer A buffer

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 Charles Kime & Thomas Kaminski 2008 Pearson Education, Inc. (Hyperlinks are active in

More information

Combinational Logic Circuits Part II -Theoretical Foundations

Combinational Logic Circuits Part II -Theoretical Foundations Combinational Logic Circuits Part II -Theoretical Foundations Overview Boolean Algebra Basic Logic Operations Basic Identities Basic Principles, Properties, and Theorems Boolean Function and Representations

More information

Unit 2 Session - 6 Combinational Logic Circuits

Unit 2 Session - 6 Combinational Logic Circuits Objectives Unit 2 Session - 6 Combinational Logic Circuits Draw 3- variable and 4- variable Karnaugh maps and use them to simplify Boolean expressions Understand don t Care Conditions Use the Product-of-Sums

More information

Standard & Canonical Forms

Standard & Canonical Forms 1 COE 202- Digital Logic Standard & Canonical Forms Dr. Abdulaziz Y. Barnawi COE Department KFUPM 2 Outline Minterms and Maxterms From truth table to Boolean expression Sum of minterms Product of Maxterms

More information

Chapter 3. Boolean Algebra. (continued)

Chapter 3. Boolean Algebra. (continued) Chapter 3. Boolean Algebra (continued) Algebraic structure consisting of: set of elements B binary operations {+, -} unary operation {'} Boolean Algebra such that the following axioms hold:. B contains

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

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

Digital Logic Design. Combinational Logic

Digital Logic Design. Combinational Logic Digital Logic Design Combinational Logic Minterms A product term is a term where literals are ANDed. Example: x y, xz, xyz, A minterm is a product term in which all variables appear exactly once, in normal

More information

Review: Additional Boolean operations

Review: Additional Boolean operations Review: Additional Boolean operations Operation: NAND (NOT-AND) NOR (NOT-OR) XOR (exclusive OR) Expressions: (xy) = x + y (x + y) = x y x y = x y + xy Truth table: x y (xy) x y (x+y) x y x y 0 0 1 0 1

More information

Goals for Lecture. Binary Logic and Gates (MK 2.1) Binary Variables. Notation Examples. Logical Operations

Goals for Lecture. Binary Logic and Gates (MK 2.1) Binary Variables. Notation Examples. Logical Operations Introduction to Electrical Engineering, II LETURE NOTES #2 Instructor: Email: Telephone: Office: ndrew. Kahng (lecture) abk@ucsd.edu 858-822-4884 office 3802 P&M lass Website: http://vlsicad.ucsd.edu/courses/ece20b/wi04/

More information

Learning Objectives 10/7/2010. CE 411 Digital System Design. Fundamental of Logic Design. Review the basic concepts of logic circuits. Dr.

Learning Objectives 10/7/2010. CE 411 Digital System Design. Fundamental of Logic Design. Review the basic concepts of logic circuits. Dr. /7/ CE 4 Digital ystem Design Dr. Arshad Aziz Fundamental of ogic Design earning Objectives Review the basic concepts of logic circuits Variables and functions Boolean algebra Minterms and materms ogic

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

CprE 281: Digital Logic

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

More information

Computer Organization I

Computer Organization I Computer Organization I Lecture 6: Boolean Algebra /2/29 Wei Lu CS283 Overview Two Principles in Boolean Algebra () Duality Principle (2) Complement Principle Standard Form of Logic Expression () Sum of

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

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 electronics form a class of circuitry where the ability of the electronics to process data is the primary focus.

Digital electronics form a class of circuitry where the ability of the electronics to process data is the primary focus. Chapter 2 Digital Electronics Objectives 1. Understand the operation of basic digital electronic devices. 2. Understand how to describe circuits which can process digital data. 3. Understand how to design

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

Signals and Systems Digital Logic System

Signals and Systems Digital Logic System Signals and Systems Digital Logic System Prof. Wonhee Kim Chapter 2 Design Process for Combinational Systems Step 1: Represent each of the inputs and outputs in binary Step 1.5: If necessary, break the

More information

COSC3330 Computer Architecture Lecture 2. Combinational Logic

COSC3330 Computer Architecture Lecture 2. Combinational Logic COSC333 Computer rchitecture Lecture 2. Combinational Logic Instructor: Weidong Shi (Larry), PhD Computer Science Department University of Houston Today Combinational Logic oolean lgebra Mux, DeMux, Decoder

More information

Lecture 5: NAND, NOR and XOR Gates, Simplification of Algebraic Expressions

Lecture 5: NAND, NOR and XOR Gates, Simplification of Algebraic Expressions EE210: Switching Systems Lecture 5: NAND, NOR and XOR Gates, Simplification of Algebraic Expressions Prof. YingLi Tian Feb. 15, 2018 Department of Electrical Engineering The City College of New York The

More information

Boolean Algebra CHAPTER 15

Boolean Algebra CHAPTER 15 CHAPTER 15 Boolean Algebra 15.1 INTRODUCTION Both sets and propositions satisfy similar laws, which are listed in Tables 1-1 and 4-1 (in Chapters 1 and 4, respectively). These laws are used to define an

More information

Chapter 7 Logic Circuits

Chapter 7 Logic Circuits Chapter 7 Logic Circuits Goal. Advantages of digital technology compared to analog technology. 2. Terminology of Digital Circuits. 3. Convert Numbers between Decimal, Binary and Other forms. 5. Binary

More information