# Boolean Algebra. BME208 Logic Circuits Yalçın İŞLER

Size: px
Start display at page:

Transcription

1 Boolean Algebra BME28 Logic Circuits Yalçın İŞLER

2 5 Boolean Algebra /2 A set of elements B There exist at least two elements x, y B s. t. x y Binary operators: + and closure w.r.t. both + and additive identity? multiplicative identity? commutative w.r.t. both + and Associative w.r.t. both + and Distributive law: is distributive over +? + is distributive over? We do not have both in ordinary algebra

3 Boolean Algebra 2/2 Complement x B, there exist an element x B a. x + x = (multiplicative identity) and b. x x = (additive identity) Not available in ordinary algebra Differences btw ordinary and Boolean algebra Ordinary algebra with real numbers Boolean algebra with elements of set B Complement Distributive law Do not substitute laws from one to another where they are not applicable 6

4 Two-Valued Boolean Algebra /3 To define a Boolean algebra The set B Rules for two binary operations The elements of B and rules should conform to our axioms Two-valued Boolean algebra B = {, } x y x y x y x + y x x 7

5 Two-Valued Boolean Algebra 2/3 Check the axioms Two distinct elements, Closure, associative, commutative, identity elements Complement x + x = and x x = Distributive law x y z 8 y z x+(y z) x + y x + z (x + y) (x + z)

6 Two-Valued Boolean Algebra 3/3 Two-valued Boolean algebra is actually equivalent to the binary logic defined heuristically before Operations: AND + OR Complement NOT Binary logic is application of Boolean algebra to the gate-type circuits Two-valued Boolean algebra is developed in a formal mathematical manner This formalism is necessary to develop theorems and properties of Boolean algebra 9

7 Duality Principle An important principle every algebraic expression deducible from the axioms of Boolean algebra remains valid if the operators and identity elements are interchanged Example: x + x = x x + x = (x+x) (identity element) = (x+x)(x+x ) (complement) = x+xx (+ over ) = x (complement) duality principle x + x = x x x = x

8 Duality Principle & Theorems Theorem a: x + = x + = (x + ) = (x + x )(x + ) = x + x =x + x = Theorem b: (using duality) x. =

9 Absorption Theorem a. x + xy = x =x.+xy =x(+y) =x 2

10 Involution & DeMorgan s Theorems Involution Theorem: (x ) = x x + x = and x x = Complement of x is x Complement is unique DeMorgan s Theorem: a. (x + y) = x y b. From duality? (x.y) =x +y 3

11 Truth Tables for DeMorgan s Theorem (x + y) = x y x y x+y (x+y) x y (x y) 4 x y x y x + y

12 Operator Precedence. Parentheses 2. NOT 3. AND 4. OR Example: (x + y) x y x + x y 5

13 Boolean Functions Consists of binary variables (normal or complement form) the constants, and logic operation symbols, + and Example: F (x, y, z) = x + y z F 2 (x, y, z) = x y z + x y z + xy x y z F F 2 6

14 Logic Circuit Diagram of F F (x, y, z) = x + y z x y z y x + y z y z Gate Implementation of F = x + y z 7

15 Logic Circuit Diagram of F 2 F 2 = x y z + x y z + xy x y z F 2 Algebraic manipulation F 2 = x y z + x y z + xy = x z(y +y) + xy 8

16 Alternative Implementation of F 2 F 2 = x z + xy x y z F 2 F 2 = x y z + x y z + xy x y z F 2 9

17 Example

18 Example

19 Example

20 OTHER LOGIC OPERATORS - AND, OR, NOT are logic operators Boolean functions with two variables three of the 6 possible two-variable Boolean functions x y F F F 2 F 3 F 4 F 5 F 6 F 7 23 x y F 8 F 9 F F F 2 F 3 F 4 F 5

21 OTHER LOGIC OPERATORS - 2 Some of the Boolean functions with two variables Constant functions: F = and F 5 = AND function: F = xy OR function: F 7 = x + y XOR function: F 6 = x y + xy = x y (x or y, but not both) XNOR (Equivalence) function: F 9 = xy + x y = (x y) (x equals y) NOR function: F 8 = (x + y) = (x y) (Not-OR) NAND function: F 4 = (x y) = (x y) (Not-AND) 24

22 Logic Gate Symbols NOT TRANSFER AND XOR OR XNOR NAND NOR 25

23 Universal Gates NAND and NOR gates are universal We know any Boolean function can be written in terms of three logic operations: AND, OR, NOT In return, NAND gate can implement these three logic gates by itself So can NOR gate x y (xy) x y (x y ) 26

24 NAND Gate x x y x y 27

25 NOR Gate x x y x y 28

26 A function: F = x y + xy Designs with NAND gates Example /2 x y 29

27 Example 2/2 F 2 = x y + xy x y 3

28 Multiple Input Gates AND and OR operations: They are both commutative and associative No problem with extending the number of inputs NAND and NOR operations: they are both commutative but not associative Extending the number of inputs is not obvious Example: NAND gates ((xy) z) (x(yz) ) ((xy) z) = xy + z (x(yz) ) = x + yz 3

29 Nonassociativity of NOR operation z 32

30 Multiple Input Universal Gates To overcome this difficulty, we define multipleinput NAND and NOR gates in slightly different manner Three input NAND gate: (x y z) x y z (x y z) Three input NOR gate:(x + y + z) x y z (x + y + z) 33

31 Multiple Input Universal Gates x y z (x + y + z) x y z (xyz) 3-input NOR gate 3-input NAND gate A B C D E F = Cascaded NAND gates 34

32 XOR and XNOR Gates XOR and XNOR operations are both commutative and associative. No problem manufacturing multiple input XOR and XNOR gates However, they are more costly from hardware point of view. Therefore, we usually have 2-input XOR and XNOR gates x y x y z z 35

33 36 3-input XOR Gates

34 38 Complement of a Function F is complement of F We can obtain F, by interchanging of s and s in the truth table F z F y x F = F =

35 Generalizing DeMorgan s Theorem We can also utilize DeMorgan s Theorem (x + y) = x y (A + B + C) = (A+B) C = A B C We can generalize DeMorgan s Theorem.(x + x x N ) = x x 2 x N 2. (x x 2 x N ) = x + x x N 39

36 Example: Complement of a Function Example: F = x yz + x y z F = (x yz + x y z) = (x yz ) (x y z) = (x + y + z)(x + y + z ) F 2 = x(y z + yz) F 2 = (x(y z + yz)) = x +(y z + yz) = x + (y + z) (y + z ) Easy Way to Complement: take the dual of the function and complement each literal 4

37 Minterms 4 Canonical & Standard Forms A product term: all variables appear (either in its normal, x, or its complement form, x ) How many different terms we can get with x and y? x y m x y m xy m 2 xy m 3 m, m, m 2, m 3 (minterms or AND terms, standard product) n variables can be combined to form 2 n minterms

38 Canonical & Standard Forms Maxterms (OR terms, standard sums) M = x + y M = x + y M 2 = x + y M 3 = x + y n variables can be combined to form 2 n maxterms m = M m = M m 2 = M 2 m 3 = M 3 42

39 Example xyz m i M i F m =x y z M =x+y+z m =x y z M =x+y+z m 2 =x yz M 2 =x+y +z m 3 =x yz M 3 =x+y +z m 4 =xy z M 4 =x +y+z m 5 =xy z M 5 =x +y+z m 6 =xyz M 6 =x +y +z m 7 =xyz M 7 =x +y +z F(x, y, z) = x y z + x yz + xyz F(x, y, z) = (x+y+z)(x+y +z )(x +y+z)(x +y+z )(x +y +z ) 43

40 Formal Expression with Minterms xyz m i M i F m =x y z M =x+y+z F(,,) m =x y z M =x+y+z F(,,) m 2 =x yz M 2 =x+y +z F(,,) m 3 =x yz M 3 =x+y +z F(,,) m 4 =xy z M 4 =x +y+z F(,,) m 5 =xy z M 5 =x +y+z F(,,) m 6 =xyz M 6 =x +y +z F(,,) m 7 =xyz M 7 =x +y +z F(,,) F(x, y, z) = F(,,)m + F(,,)m + F(,,)m 2 + F(,,)m 3 + F(,,)m 4 + F(,,)m 5 + F(,,)m 6 + F(,,)m 7 45

41 Formal Expression with Maxterms xyz m i M i F m =x y z M =x+y+z F(,,) m =x y z M =x+y+z F(,,) m 2 =x yz M 2 =x+y +z F(,,) m 3 =x yz M 3 =x+y +z F(,,) m 4 =xy z M 4 =x +y+z F(,,) m 5 =xy z M 5 =x +y+z F(,,) m 6 =xyz M 6 =x +y +z F(,,) m 7 =xyz M 7 =x +y +z F(,,) F(x, y, z) = (F(,,)+M ) (F(,,)+M ) (F(,,)+M 2 ) (F(,,)+M 3 ) (F(,,)+M 4 ) (F(,,)+M 5 ) (F(,,)+M 6 ) (F(,,)+M 7 ) 46

42 Boolean Functions in Standard Form F (x, y, z) = F 2 (x, y, z) = 47 x y z F F 2

43 Important Properties Any Boolean function can be expressed as a sum of minterms Any Boolean function can be expressed as a product of maxterms Example: F = (, 2, 3, 5, 6) = x y z + x yz + x yz + xy z + xyz How do we find the complement of F? F = (x + y + z)(x + y + z)(x + y + z )(x + y + z )(x + y + z) = = 48

44 Canonical Form If a Boolean function is expressed as a sum of minterms or product of maxterms the function is said to be in canonical form. Example: F = x + y z canonical form? No But we can put it in canonical form. F = x + y z = (7, 6, 5, 4, ) Alternative way: Obtain the truth table first and then the canonical term. 49

45 Example: Product of Maxterms F = xy + x z Use the distributive law + over F = xy + x z = xy(z+z ) + x z(y+y ) =xyz + xyz +x yz + x y z = (7,6,3,) 5 = (4, 5,, 2)

46 Conversion Between Canonical Forms Fact: The complement of a function (given in sum of minterms) can be expressed as a sum of minterms missing from the original function Example: F(x, y, z) = (, 4, 5, 6, 7) F (x, y, z) = Now take the complement of F and make use of DeMorgan s theorem (F ) = = F = M M 2 M 3 = (, 2, 3) 5

47 General Rule for Conversion Important relation: m j = M j. M j = m j. The rule: Interchange symbols and, and list those terms missing from the original form Example: F = xy + x z F = (, 3, 6, 7) F = (?,?,?,?) 52

48 53 Standard Forms Fact: Canonical forms are very seldom the ones with the least number of literals Alternative representation: Standard form a term may contain any number of literals Two types. the sum of products 2. the product of sums Examples: F = y + xy + x yz F 2 = x(y + z)(x + y + z )

49 Example: Standard Forms F = y + xy + x yz F 2 = x(y + z)(x + y + z ) 54

50 Example: F 3 = AB(C+D) + C(D + E) 55 Nonstandard Forms This hybrid form yields three-level implementation A B C D D E C The standard form: F 3 = ABC + ABD + CD + CE A B C A B D C D C E F 3 F 3

51 Positive & Negative Logic In digital circuits, we have two digital signal levels: H (higher signal level; e.g. 3 ~ 5 V) L - (lower signal level; e.g. ~ V) There is no logic- or logic- at the circuit level We can do any assignment we wish For example: H logic- L logic- 56

52 Feedforward or Feedback x 4 x x 2 x4 x 3 x 5 57

53 Feedforward or Feedback x 4 x x 2 z y x4 x 3 x 5 z = x x 2 (x 4 +x 3 x 5 (x 4 +y)) 58

54 Feedforward or Feedback x 4 x x 2 z y x4 x 3 x 5 z = x x 2 (x 4 +x 3 x 5 (x 4 +y))=x x 2 (x 4 +x 3 x 5 x 4 +x 3 x 5 y) 59

55 Feedforward or Feedback x 4 x x 2 z y x4 x 3 x 5 6 z = x x 2 (x 4 +x 3 x 5 (x 4 +y))=x x 2 (x 4 +x 3 x 5 x 4 +x 3 x 5 y) = x x 2 (x 4 +x 3 x 5 +x 3 x 5 y)

56 Feedforward or Feedback x 4 x x 2 z y x4 x 3 x 5 6 z = x x 2 (x 4 +x 3 x 5 (x 4 +y))=x x 2 (x 4 +x 3 x 5 x 4 +x 3 x 5 y) = x x 2 (x 4 +x 3 x 5 +x 3 x 5 y)=x x 2 (x 4 +x 3 x 5 (+y))

57 Feedforward or Feedback x 4 x x 2 z y x4 x 3 x 5 62 z = x x 2 (x 4 +x 3 x 5 (x 4 +y))=x x 2 (x 4 +x 3 x 5 x 4 +x 3 x 5 y) = x x 2 (x 4 +x 3 x 5 +x 3 x 5 y)=x x 2 (x 4 +x 3 x 5 )

58 Signal Designation - x y digital gate F x y F L L L L H H H L H H H H What kind of logic function does it implement? 63

59 Signal Designation - 2 x y F x y F x y x y positive logic negative logic F F polarity indicator 64

60 Another Example x y F L L H L H H H L H H H L 74LS 65

61 Integrated Circuits IC silicon semiconductor crystal ( chip ) that contains gates. gates are interconnected inside to implement a Boolean function Chip is mounted in a ceramic or plastic container Inputs & outputs are connected to the external pins of the IC. Many external pins (4 to hundreds) 66

62 Levels of Integration SSI (small-scale integration): Up to gates per chip MSI (medium-scale integration): From to, gates per chip LSI (large-scale integration): thousands of gates per chip VLSI (very large-scale integration): hundreds of thousands of gates per chip ULSI (ultra large-scale integration): Over a million gates per chip 94 million transistors 67

63 Digital Logic Families Circuit Technologies TTL transistor-transistor logic ECL Emitter-coupled logic fast MOS metal-oxide semiconductor high density CMOS Complementary MOS low power 68

64 Parameters of Logic Gates - Fan-out load that the output of a gate drives If a gate, say NAND, drives four such inverters, then the fan-out is equal to 4. standard loads. 69

65 Parameters of Logic Gates -

66 Parameters of Logic Gates - 2 Fan-in number of inputs that a gate can have in a particular logic family In principle, we can design a CMOS NAND or NOR gate with a very large number of inputs In practice, however, we have some limits 4 for NOR gates 6 for NAND gates Power dissipation power consumed by the gate that must be available from the power supply 7

67 Power Dissipation Difference in voltage level! 72

68 Parameters of Logic Gates - 3 Propagation delay: the time required for a change in value of a signal to propagate from input to output. logic- logic- logic- x y logic- F logic- t PHL t PLH x time y F 73

69 Computer-Aided Design - CAD Design of digital systems with VLSI circuits containing millions of transistors is not easy and cannot be done manually. To develop & verify digital systems we need CAD tools software programs that support computer-based representation of digital circuits. Design process design entry database that contains the photomask used to fabricate the IC 74

70 Computer-Aided Design - 2 Different physical realizations ASIC (application specific integrated circuit) PLD FPGA Other reconfigurable devices For every piece of device we have an array of software tools to facilitate design, simulating, testing, and even programming 75

71 76 Xilinx Tools

72 Schematic Editor Editing programs for creating and modifying schematic diagrams on a computer screen schematic capturing or schematic entry you can drag-and-drop digital components from a list in an internal library (gates, decoders, multiplexers, registers, etc.) You can draw interconnect lines from one component to another 77

73 78 Schematic Editor

74 79 A Schematic Design

75 Hardware Description Languages HDL Verilog, VHDL resembles a programming language designed to describe digital circuits so that we can develop and test digital circuits module comp(f, x, y, z, t); input x, y, z, t; output F; F (x,y,z,t) = xz + yz t + xyt 8 wire e, e2, e3; and g(e, x, ~z); and g2(e2, y, ~z, ~t); and g3(e3, x, y, ~t); or g4(f, e, e2, e3); endmodule

76 8 Simulation Results /3

77 82 Simulation Results 2/3

78 83 Simulation Results 3/3

### 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

### 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 Tong In Oh 1 Basic Definitions 2 3 2.3 Axiomatic Definition of Boolean Algebra Boolean algebra: Algebraic structure defined by a set of elements, B, together

### 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

### Chap-2 Boolean Algebra

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

### 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

### Gate-Level Minimization. BME208 Logic Circuits Yalçın İŞLER

Gate-Level Minimization BME28 Logic Circuits Yalçın İŞLER islerya@yahoo.com http://me.islerya.com Complexity of Digital Circuits Directly related to the complexity of the algebraic expression we use to

### 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

### 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

### 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 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

### 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

### 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

### Software Engineering 2DA4. Slides 2: Introduction to Logic Circuits

Software Engineering 2DA4 Slides 2: Introduction to Logic Circuits Dr. Ryan Leduc Department of Computing and Software McMaster University Material based on S. Brown and Z. Vranesic, Fundamentals of Digital

### 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

### 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

### 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-

### 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)

### CONTENTS CHAPTER 1: NUMBER SYSTEM. Foreword...(vii) Preface... (ix) Acknowledgement... (xi) About the Author...(xxiii)

CONTENTS Foreword...(vii) Preface... (ix) Acknowledgement... (xi) About the Author...(xxiii) CHAPTER 1: NUMBER SYSTEM 1.1 Digital Electronics... 1 1.1.1 Introduction... 1 1.1.2 Advantages of Digital Systems...

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

### 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

### 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?

### 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

### 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

### 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.

### 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

### 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

### 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.

### 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)

### 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

Get Free notes at Module-I One s Complement: Complement all the bits.i.e. makes all 1s as 0s and all 0s as 1s Two s Complement: One s complement+1 SIGNED BINARY NUMBERS Positive integers (including zero)

### 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

### 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

### 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 =

### 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

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

L T P C R.M.D. ENGINEERING COLLEGE R.S.M. Nagar, Kavaraipettai- 601206 DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING EC8392 UNIT - I 3 0 0 3 OBJECTIVES: To present the Digital fundamentals, Boolean

### 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

### Circuit analysis summary

Boolean Algebra Circuit analysis summary After finding the circuit inputs and outputs, you can come up with either an expression or a truth table to describe what the circuit does. You can easily convert

### 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

### Gate-Level Minimization

Gate-Level Minimization ( 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 The Map Method

### 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 ):

### 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

### 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

### 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

### Chapter 3. Gate-Level Minimization. Outlines

Chapter 3 Gate-Level Minimization Introduction The Map Method Four-Variable Map Five-Variable Map Outlines Product of Sums Simplification Don t-care Conditions NAND and NOR Implementation Other Two-Level

### 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:

### 數位系統 Digital Systems 朝陽科技大學資工系. Speaker: Fuw-Yi Yang 楊伏夷. 伏夷非征番, 道德經察政章 (Chapter 58) 伏者潛藏也道紀章 (Chapter 14) 道無形象, 視之不可見者曰夷

數位系統 Digital Systems Department of Computer Science and Information Engineering, Chaoyang University of Technology 朝陽科技大學資工系 Speaker: Fuw-Yi Yang 楊伏夷 伏夷非征番, 道德經察政章 (Chapter 58) 伏者潛藏也道紀章 (Chapter 14) 道無形象,

### 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

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

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

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

### This presentation will..

Component Identification: Digital Introduction to Logic Gates and Integrated Circuits Digital Electronics 2014 This presentation will.. Introduce transistors, logic gates, integrated circuits (ICs), and

### 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

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

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

### DIGITAL ELECTRONICS. Vayu Education of India

DIGITAL ELECTRONICS ARUN RANA Assistant Professor Department of Electronics & Communication Engineering Doon Valley Institute of Engineering & Technology Karnal, Haryana (An ISO 9001:2008 ) Vayu Education

### Digital Logic Design. Outline

Digital Logic Design Gate-Level Minimization CSE32 Fall 2 Outline The Map Method 2,3,4 variable maps 5 and 6 variable maps (very briefly) Product of sums simplification Don t Care conditions NAND and NOR

### 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,

### DIGITAL CIRCUIT LOGIC UNIT 9: MULTIPLEXERS, DECODERS, AND PROGRAMMABLE LOGIC DEVICES

DIGITAL CIRCUIT LOGIC UNIT 9: MULTIPLEXERS, DECODERS, AND PROGRAMMABLE LOGIC DEVICES 1 Learning Objectives 1. Explain the function of a multiplexer. Implement a multiplexer using gates. 2. Explain the

### Gate-Level Minimization. section instructor: Ufuk Çelikcan

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

### 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

### 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

### 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

### 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

### Henry Lin, Department of Electrical and Computer Engineering, California State University, Bakersfield Lecture 7 (Digital Logic) July 24 th, 2012

Henry Lin, Department of Electrical and Computer Engineering, California State University, Bakersfield Lecture 7 (Digital Logic) July 24 th, 2012 1 Digital vs Analog Digital signals are binary; analog

### Hours / 100 Marks Seat No.

17333 13141 3 Hours / 100 Seat No. Instructions (1) All Questions are Compulsory. (2) Answer each next main Question on a new page. (3) Illustrate your answers with neat sketches wherever necessary. (4)

### 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

### 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

### VALLIAMMAI ENGINEERING COLLEGE. SRM Nagar, Kattankulathur DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING EC6302 DIGITAL ELECTRONICS

VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur-603 203 DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING EC6302 DIGITAL ELECTRONICS YEAR / SEMESTER: II / III ACADEMIC YEAR: 2015-2016 (ODD

### Programmable Logic Devices (PLDs)

Programmable Logic Devices (PLDs) 212: Digital Design I, week 13 PLDs basically store binary information in a volatile/nonvolatile device. Data is specified by designer and physically inserted (Programmed)

### Purdue IMPACT 2015 Edition by D. G. Meyer. Introduction to Digital System Design. Module 2 Combinational Logic Circuits

Purdue IMPACT 25 Edition by D. G. Meyer Introduction to Digital System Design Module 2 Combinational Logic Circuits Glossary of Common Terms DISCRETE LOGIC a circuit constructed using small-scale integrated

### Gate-Level Minimization

Gate-Level Minimization ( 范倫達 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C. Fall, 2011 ldvan@cs.nctu.edu.tw http://www.cs.nctu.edu.tw/~ldvan/ Outlines The Map Method

### 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

### 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

### Code No: R Set No. 1

Code No: R059210504 Set No. 1 II B.Tech I Semester Regular Examinations, November 2006 DIGITAL LOGIC DESIGN ( Common to Computer Science & Engineering, Information Technology and Computer Science & Systems

### 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

### 3. The high voltage level of a digital signal in positive logic is : a) 1 b) 0 c) either 1 or 0

1. The number of level in a digital signal is: a) one b) two c) four d) ten 2. A pure sine wave is : a) a digital signal b) analog signal c) can be digital or analog signal d) neither digital nor analog

### 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

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

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

### Intro to Logic Gates & Datasheets. Intro to Logic Gates & Datasheets. Introduction to Integrated Circuits. TTL Vs. CMOS Logic

Intro to Logic Gates & Datasheets Digital Electronics Intro to Logic Gates & Datasheets This presentation will Introduce integrated circuits (ICs). Present an overview of : Transistor-Transistor Logic

### Intro to Logic Gates & Datasheets. Digital Electronics

Intro to Logic Gates & Datasheets Digital Electronics Intro to Logic Gates & Datasheets This presentation will Introduce integrated circuits (ICs). Present an overview of : Transistor-Transistor Logic

### 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

### 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

### Combinational Logic Use the Boolean Algebra and the minimization techniques to design useful circuits No feedback, no memory Just n inputs, m outputs

Combinational Logic Use the Boolean Algebra and the minimization techniques to design useful circuits No feedback, no memory Just n inputs, m outputs and an arbitrary truth table Analysis Procedure We

### 211: Computer Architecture Summer 2016

211: Computer Architecture Summer 2016 Liu Liu Topic: Storage Project3 Digital Logic - Storage: Recap - Direct - Mapping - Fully Associated - 2-way Associated - Cache Friendly Code Rutgers University Liu

### 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.

### 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)

### Combinational Logic. Prof. Wangrok Oh. Dept. of Information Communications Eng. Chungnam National University. Prof. Wangrok Oh(CNU) 1 / 93

Combinational Logic Prof. Wangrok Oh Dept. of Information Communications Eng. Chungnam National University Prof. Wangrok Oh(CNU) / 93 Overview Introduction 2 Combinational Circuits 3 Analysis Procedure

### Introduction to Programmable Logic Devices (Class 7.2 2/28/2013)

Introduction to Programmable Logic Devices (Class 7.2 2/28/2013) CSE 2441 Introduction to Digital Logic Spring 2013 Instructor Bill Carroll, Professor of CSE Today s Topics Complexity issues Implementation

### 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

### Chap.3 3. Chap reduces the complexity required to represent the schematic diagram of a circuit Library

3.1 Combinational Circuits 2 Chap 3. logic circuits for digital systems: combinational vs sequential Combinational Logic Design Combinational Circuit (Chap 3) outputs are determined by the present applied

### Computer Organization

Computer Organization (Logic circuits design and minimization) KR Chowdhary Professor & Head Email: kr.chowdhary@gmail.com webpage: krchowdhary.com Department of Computer Science and Engineering MBM Engineering

### ENGIN 112 Intro to Electrical and Computer Engineering

ENGIN 2 Intro to Electrical and Computer Engineering Lecture 5 Boolean Algebra Overview Logic functions with s and s Building digital circuitry Truth tables Logic symbols and waveforms Boolean algebra

### 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

### Combinational Logic II

Combinational Logic II Ranga Rodrigo July 26, 2009 1 Binary Adder-Subtractor Digital computers perform variety of information processing tasks. Among the functions encountered are the various arithmetic

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

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

### DIGITAL DESIGN TECHNOLOGY & TECHNIQUES

DIGITAL DESIGN TECHNOLOGY & TECHNIQUES CAD for ASIC Design 1 INTEGRATED CIRCUITS (IC) An integrated circuit (IC) consists complex electronic circuitries and their interconnections. William Shockley et

### 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

### SUBJECT CODE: IT T35 DIGITAL SYSTEM DESIGN YEAR / SEM : 2 / 3

UNIT - I PART A (2 Marks) 1. Using Demorgan s theorem convert the following Boolean expression to an equivalent expression that has only OR and complement operations. Show the function can be implemented