# Boolean Algebra & Digital Logic

Size: px
Start display at page:

Transcription

1 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 Found the Mathematical Theories of Logic and Probabilities. The applicability to computing machines was discovered by three Americans Claude Shannon Symbolic Analysis of Relay and Switching Circuits, George Stibitz John Atanasoff An employee of Bell Labs, he developed a binary adder using mechanical relays in 1937, the model K 1 adder because he built it at home on his kitchen table. He was probably the first to use purely electronic relays (vacuum tubes) to build a binary adder. Boolean algebra is a two valued algebra based on the constant values denoted as either FALSE, TRUE 0, 1 The use of this algebra for computation is based on the fact that binary arithmetic is based on two values, always called 0 and 1.

2 Basic Boolean Operators Boolean algebra is defined in terms of two constants (defined above), which we call 0 and 1. Other courses will call these values F and T. Boolean algebra is defined in terms of three basic operators, to which we shall add a useful fourth operator. The three operators are NOT, AND, & OR. Each of these three basic operators is implemented by a basic electronic device called a logic gate. We present the gates along with the definition. NOT This function takes one input and produces one output. The gate is shown below. The circle at the right end of the triangle is important. Algebraically, this function is denoted f(x) = X or f(x) = X The evaluation of the function is simple: 0= 1 and 1 = 0.

3 Basic Boolean Operators (Part 2) Logic OR This is a function of two Boolean variables. We denote the logical OR of two Boolean variables X and Y by X + Y. Some logic books will use X Y. The evaluation of the logical OR function is shown by a truth table X Y X + Y

4 Basic Boolean Operators (Part 3) Logic AND This is a function of two Boolean variables. We denote the logical AND of two Boolean variables X and Y by X Y. Some logic books will use X Y. The evaluation of the logical AND function is shown by a truth table X Y X Y

5 Another Boolean Operator While not a basic Boolean operator, the exclusive OR is very handy. Logic XOR This is a function of two Boolean variables. We denote the logical XOR of two Boolean variables X and Y by X Y. Most logic books seem to ignore this function. The evaluation of the logical XOR function is shown by a truth table X Y X Y From this last table, we see immediately that X 0 = X and X 1 = X

6 Time Varying Inputs and Outputs Consider the figure below, adapted from the text by Rob Williams. What does it tell us? This is a two input AND gate, with inputs D (Data) and X (Control). If X = 0, the output will be 0 independently of the value of the data input. The simple answer is that the gate output follows the input when X = 1. The input is a time sequence of values. At any time, the data input to the AND gate is either d = 0 or d = 1. At that time, the output from the gate is either Y = 0 or Y = 1. The use of the letter O as in O = 1 is a bit hard to read; it appears confused.

7 More on the Time Sequence

8 Truth Tables The fact that any Boolean variable can assume only one of two possibly values can be shown, by induction, to imply the following. For N > 0, N Boolean variables can take only 2 N different combinations of values. For small values of N, we can use this to specify a function using a truth table with 2 N rows, plus a header row to label the variables and the function. N Four variable truth tables have 17 rows total. This is just manageable. Five variable truth tables have 33 rows total. This is excessive. N variable truth tables, for N > 5, are almost useless. 2 N

9 Sample Truth Table A B C F1(A, B, C) This truth table for 3 variables has 2 3 = 8 rows, plus a label row. This truth table forms a complete definition of the function. We shall later give it another name, but can base all our discussions on this table.

10 Another Sample Truth Table A B C F2(A, B, C)

11 Two Truth Tables in One A B C F1(A, B, C) F2(A, B, C) Truth tables are often used to show pairs of functions, such as these two, which will later be shown to be related. This is easier than two complete tables. Truth tables rarely show more than two functions, just because large truth tables are messy and hard to read.

12 Labeling Rows in a Truth Table The row numbers are just labels. They are not really a part of the truth table, but aid in our discussions and conversions to Boolean expressions. The row numbers are the decimal equivalents of the variable values viewed as binary Row Number X Y Z F(X, Y, Z) numbers. The first row is always row 0. 0 = = = = = = = =

13 The relative precedence of the operators is: 1) NOT do this first 2) AND 3) OR do this last Evaluation of Boolean Expressions As in the usual algebra, parentheses take precedence. A B + C D, often written as AB + CD, is read as (A B) + (C D) A B C D is read as A B C D A B + C D = = = 1 A (B + C) D = 1 (0 + 1) 1 = = 1 A B C D = A B = 1 0 = 0 = 1 A B = 1 0 = 0 1 = 0. The latter is really messy. = = = 0

14 The Basic Unusual Boolean Theorem Here are two sets of theorems in Boolean algebra. For all X 0 X = 0 OK For all X 1 X = X OK For all X 0 + X = X OK For all X 1 + X = 1 What? Consider the following truth tables W X W + X From this, we derive the truth table proving the last two theorems. X 0 + X 1 + X

15 Standard Boolean Forms In this section, we develop the idea of standard forms of Boolean expressions. In part, these forms are based on some standard Boolean simplification rules. Standard forms are either canonical forms or normal forms. The standard expressions are in either SOP POS Sum of Products form, or Product of Sums form. This lecture will focus on the following: Canonical Sum of Products Normal Sum of Products Canonical Product of Sums Normal Product of Sums We shall also discuss a few more variants that have no standard names. IMPORTANT: These forms use only the 3 basic Boolean functions: AND, OR, NOT. Specifically, XOR is not used.

16 Variables and Literals We start with the idea of a Boolean variable. It is a simple variable that can take one of only two values: 0 (False) or 1 (True). Following standard digital design practice, we use the values 0 and 1. Following standard teaching practice, we denote all Boolean variables by single letters; normally A, B, C, D, or W, X, Y, Z. A literal is either a Boolean variable or its complement. Literals based on the variable X: X and X. Literals based on the variable Y: Y and Y. NOTE: X and X represent the same variable, but they are not the same literal. X and Y represent different variables.

17 Product and Sum Terms A product term is the logical AND of one or more literals, with no variable represented more than once. A sum term is the logical OR of one or more literals, with no variable represented more than once. The following are all valid product terms over the two variables X and Y. X Y X Y X Y X Y Forms, such as X X Y and X X Y are not considered, as X X = X and X X = 0, so X X Y = X Y and X X Y = 0 Y = 0. The following are all valid sum terms over the two variables X and Y. X + Y X + Y X + Y X + Y Single literals According to the strict definition, a single literal is either a sum term or a product term, depending on the context. This is necessary to avoid having to give a number of special cases in the following definitions.

18 Sum of Products and Product of Sums A SOP (Sum of Products) expression is the logical OR of product terms. A POS (Product of Sums) expression is the logical AND of sum terms. Sample SOP expressions F1(X, Y) = X Y + X Y G1(X, Y) = X Y + X Y H1(X, Y, Z) = X + Y Z Note: If we did not allow single literals to be product terms, we would have trouble classifying H(X, Y, Z), which is clearly SOP. Sample POS expressions F2(X, Y) = (X+Y) ( X +Y ) G2(X, Y) = ( X +Y) (X+Y ) H2(X, Y, Z) = X (Y + Z) Note: POS expressions almost always have parentheses to indicate the correct evaluation.

19 More on Ambiguous Forms What is the form of the expression F(X, Y) = X + Y 1. SOP It is the logical OR of two product terms. Each product term is a single literal. 2. POS It is a single sum term (X + Y) Both statements are true. In general, questions such as this do not concern us. If you are asked a question like this on a test, either answer will be accepted. This ambiguity comes from the definitional necessity of mentioning the logical AND of one or more terms and the logical OR of one or more terms. With two equally good answers to an ambiguous form, pick the one you like.

20 Inclusion A product term T 1 is included in a product term T 2 if every literal in T 1 is also in T 2. A sum term T 1 is included in a sum term T 2 if every literal in T 1 is also in T 2. Consider the following examples: F(A, B, C) = A B + A C + A B C Each of A B and A C is included in A B C. G(A, B, C) = (A + B) (A + C) (A + B + C) Each of (A + B) and (A + C) is included in (A + B + C). There is no inclusion in the next expression F(A, B, C) = A B + A C + A B C The literal A does no appear in the third term. The inclusion rule is based on literals, not just variables.

21 Consider and More on Inclusion F1(A, B, C) = A B + A C + A B C F2(A, B, C) = A B + A C We claim that the two are equal for every value of A, B, and C. Let A = 0 Clearly F1(A, B, C) = 0 F2(A, B, C) = 0 Let A = 1 Then and F1(A, B, C) = B + C + B C F2(A, B, C) = B + C Notice that we still have inclusion in F1, as each of B and C is included in B C. We prove these versions of F1(A, B, C) = F2(A, B, C) using a truth table. B C B C B + C B + C + B C

22 Last Word on Inclusion If a SOP or POS expression has included terms, it can be simplified. F1(A, B, C) = A B + A C + A B C is identically equal to F2(A, B, C) = A B + A C G1(A, B, C) = (A + B) (A + C) (A + B + C) is identically equal to G2(A, B, C) = (A + B) (A + C) The conclusion is that Boolean expressions with included terms are needlessly complicated. We can simplify them by the application of trivial rules. Note that duplication is a form of inclusion. The expression F3(A, B) = A B + A B has 2 terms, each included in the other.

23 Non Standard Expressions Not every useful Boolean expression is in a standard form. F(X, Y) = X Y is not a standard form, due to the exclusive OR. G(X, Y) = X Y + (X + Y) ( X +Y) is not in a standard form. This has both a product term and a sum term. The fact that G(X, Y) can easily be converted to a standard form does not make it already in a standard form. Let s convert this to SOP. I usually have difficulty in conversion to POS, unless I am using a method I have yet to describe. The term X Y is already a product term, so we convert (X + Y) ( X +Y) to SOP. (X + Y) ( X +Y) = X ( X +Y) + Y ( X +Y) = X X + X Y + Y X + Y Y = 0 + X Y+ X Y + Y = (X + X ) Y + Y = 1 Y + Y = Y So G(X, Y) = X Y + Y = Y G(X, Y) = Y.

24 More on Non Standard Forms Look at the Boolean function G(X, Y) = X Y + Y. There are two ways to at look at this. Try both ways. Let G(X, Y) = X Y + Y = X Y + 1 Y = (X + 1) Y = 1 Y = Y Do a truth table proof of the equality. X Y X Y X Y + Y Note that the column marked X Y matches the one marked X Y + Y. The two functions are identical. Also note that the term Y is included in the term X Y, so that, by inclusión, the term X Y can be eliminated from the expression.

25 Normal and Canonical Forms A Normal SOP expression is a Sum of Products expression with no included product terms. A Normal POS expression is a Product of Sums expression with no included sum terms. A Canonical SOP expression over a set of Boolean variables is a Normal SOP expression in which each product term contains a literal for each of the Boolean variables. A Canonical POS expression over a set of Boolean variables is a Normal POS expression in which each sum term contains a literal for each of the Boolean variables. Note: Note: A canonical expression on N Boolean variables is made up of terms, each of which has exactly N literals. One can do digital design based on either normal or canonical forms. The choice usually depends on the technology used.

26 Equivalence of Canonical Forms and Truth Tables We can directly translate between either of the canonical forms and a truth table using a standard set of rules. To produce the Sum of Products representation from a truth table, a) Generate a product term for each row where the value of the function is 1. b) The variable is complemented if its value in the row is 0, otherwise it is not. To produce the Product of Sums representation from a truth table, a) Generate a sum term for each row where the value of the function is 0. b) The variable is complemented if its value in the row is 1, otherwise it is not. SOP: Row X Y X Y Terms for rows 1 and 2. Row 1: X Y, Row 2: X Y F = X Y + X Y POS: Terms for rows 0 and 3. Row 0: (X + Y), Row 3: ( X + Y ) F = (X + Y) ( X + Y )

27 SOP Example: Truth Table to Canonical Form To produce the Sum of Products representation from a truth table, a) Generate a product term for each row where the value of the function is 1. b) The variable is complemented if its value in the row is 0, otherwise it is not. Here again is the truth table. Row A B C F The term is A B C The term is A B C The term is A B C The term is A B C F2(A, B, C) = A B C + A B C + A B C + A B C

28 Example: Interpretation of a Digital Circuit Here is a sample problem, taken from the textbook The Essentials of Computer Organization and Architecture by Linda Null and Julia Lobur. The task is to represent this circuit by both a Boolean expression and a Truth Table. Admittedly, this will prove to be a silly circuit.

29 Interpreting a Digital Circuit: Step 1 Label the circuit elements (I have chosen to use numbers) and label the output of each element. Note that we are slowly building a Boolean expression. The outputs of each gate are as follows: The output of gate 1 is (X + Y), The output of gate 2 is (Y Z), The output of gate 3 is X, The output of gate 4 is X + (Y Z), and The output of gate 5 is (X + Y) [X + (Y Z)]

30 Interpreting a Digital Circuit: Step 2 For a circuit of this complexity, the best next step is to make a Truth Table. X Y Z X + Y (Y Z) X X + (Y Z) (X + Y) [X +(Y Z)] We have now solved the problem. I want to continue and produce a simpler expression. (At least I think that it is simpler).

31 Interpreting a Digital Circuit: Step 3 Present the truth table without the intermediate expressions. Use the standard rules to convert the truth table to Canonical SOP. X Y Z F(X, Y, Z) F(X, Y, Z) = X Y Z + X Y Z + X Y Z + X Y Z F(X, Y, Z) = X Y Z + X Y Z + X Y Z + X Y Z + X Y Z = X Y (Z + Z) + (X + X ) Y Z + X Y Z = X Y + Y Z + X Y Z

32 Building a Digital Circuit for a Boolean Expression We take as examples two representations of the same Boolean expression. Sum of Products SOP One OR gate connecting the output of a number of AND gates.

33 Building a Digital Circuit (Part 2) Product of Sums POS One AND gate connecting the output of a number of OR gates. There are simpler Boolean expressions that are equivalent to both F2 and G2, which are equivalent to each other. We study simplification later.

34 The Tri State Buffer Some time ago, we considered relays as automatic switches. The tri state buffer is also an automatic switch. Here are the diagrams for two of the four most popular tri state buffers. An enabled low buffer is the same as an enabled high buffer with a NOT gate. What does a tri state buffer do when it is enabled? What does a tri state buffer do when it is not enabled? What is this third state implied by the name tri state?

35 An Enabled High Tri State Buffer Consider an enabled high tri state buffer, with the enable signal called C. When C = 1, the buffer is enabled. When C = 0, the buffer is not enabled. What does the buffer do? The buffer should be considered a switch. When C = 0, there is no connection between the input A and the output F. When C = 1, the output F is connected to the input A via what appears to be a non inverting buffer. Strictly speaking, when C = 0 the output F remains connected to input A, but through a circuit that offers very high resistance to the flow of electricity. For this reason, the state is often called high impedance, impedance being an engineer s word for resistance.

36 Sample Use of Tri State Buffers Here is a circuit that uses a pair of tri state buffers to connect exactly one of two inputs to an output. The effect of the circuit is at right. Here is the equivalent circuit using the standard gates.

37 What is This Third State? The following circuits show the effect of two tri states. Here we see two switches, either of which can illuminate the light. The analogy is not exact, but the point is valid: neither switch is attempting to assert zero volts.

38 Tristate Buffers: Defining Voltages Consider the following diagram in the situation when the control is low; C = 0. What is the voltage at the output of tristate buffer I? The voltage is not determined by that buffer, and is independent of A. As this point has a direct connection to the output of tristate buffer II, which is active, the voltage at that point is determined by that tristate. When many tristate buffers output to a common circuit line, at most one of them should be active at any time. Two or more active is an error.

### Standard Boolean Forms

Standard Boolean Forms In this section, we develop the idea of standard forms of Boolean expressions. In part, these forms are based on some standard Boolean simplification rules. Standard forms are either

### 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

### 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

### 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.

### Bawar Abid Abdalla. Assistant Lecturer Software Engineering Department Koya University

Logic Design First Stage Lecture No.6 Boolean Algebra Bawar Abid Abdalla Assistant Lecturer Software Engineering Department Koya University Outlines Boolean Operations Laws of Boolean Algebra Rules of

### Chapter 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

### 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

### (Refer Slide Time 6:48)

Digital Circuits and Systems Prof. S. Srinivasan Department of Electrical Engineering Indian Institute of Technology Madras Lecture - 8 Karnaugh Map Minimization using Maxterms We have been taking about

### Announcements. Chapter 2 - Part 1 1

Announcements If you haven t shown the grader your proof of prerequisite, please do so by 11:59 pm on 09/05/2018 (Wednesday). I will drop students that do not show us the prerequisite proof after this

### 1. Mark the correct statement(s)

1. Mark the correct statement(s) 1.1 A theorem in Boolean algebra: a) Can easily be proved by e.g. logic induction b) Is a logical statement that is assumed to be true, c) Can be contradicted by another

### BOOLEAN ALGEBRA. Logic circuit: 1. From logic circuit to Boolean expression. Derive the Boolean expression for the following circuits.

COURSE / CODE DIGITAL SYSTEMS FUNDAMENTAL (ECE 421) DIGITAL ELECTRONICS FUNDAMENTAL (ECE 422) BOOLEAN ALGEBRA Boolean Logic Boolean logic is a complete system for logical operations. It is used in countless

### 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

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)

### 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

### (Refer Slide Time 3:31)

Digital Circuits and Systems Prof. S. Srinivasan Department of Electrical Engineering Indian Institute of Technology Madras Lecture - 5 Logic Simplification In the last lecture we talked about logic functions

### Module -7. Karnaugh Maps

1 Module -7 Karnaugh Maps 1. Introduction 2. Canonical and Standard forms 2.1 Minterms 2.2 Maxterms 2.3 Canonical Sum of Product or Sum-of-Minterms (SOM) 2.4 Canonical product of sum or Product-of-Maxterms(POM)

### 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

### LAB #1 BASIC DIGITAL CIRCUIT

LAB #1 BASIC DIGITAL CIRCUIT OBJECTIVES 1. To study the operation of basic logic gates. 2. To build a logic circuit from Boolean expressions. 3. To introduce some basic concepts and laboratory techniques

### Chapter 2 Combinational Logic Circuits

Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization Overview Part Gate Circuits and Boolean Equations Binary Logic and Gates Boolean Algebra Standard

### 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. 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

### BOOLEAN ALGEBRA AND CIRCUITS

UNIT 3 Structure BOOLEAN ALGEBRA AND CIRCUITS Boolean Algebra and 3. Introduction 3. Objectives 3.2 Boolean Algebras 3.3 Logic 3.4 Boolean Functions 3.5 Summary 3.6 Solutions/ Answers 3. INTRODUCTION This

### Designing Computer Systems Boolean Algebra

Designing Computer Systems Boolean Algebra 08:34:45 PM 4 June 2013 BA-1 Scott & Linda Wills Designing Computer Systems Boolean Algebra Programmable computers can exhibit amazing complexity and generality.

### Chapter 2 Combinational

Computer Engineering 1 (ECE290) Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization HOANG Trang 2008 Pearson Education, Inc. Overview Part 1 Gate Circuits and Boolean Equations Binary Logic

### Dr. Chuck Cartledge. 10 June 2015

Miscellanea Exam #1 Break Exam review 2.1 2.2 2.3 2.4 Break 3 4 Conclusion References CSC-205 Computer Organization Lecture #003 Chapter 2, Sections 2.1 through 4 Dr. Chuck Cartledge 10 June 2015 1/30

### 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:

### Introduction to Computer Architecture

Boolean Operators The Boolean operators AND and OR are binary infix operators (that is, they take two arguments, and the operator appears between them.) A AND B D OR E We will form Boolean Functions of

### Introduction to Boolean Algebra

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

### LSN 4 Boolean Algebra & Logic Simplification. ECT 224 Digital Computer Fundamentals. Department of Engineering Technology

LSN 4 Boolean Algebra & Logic Simplification Department of Engineering Technology LSN 4 Key Terms Variable: a symbol used to represent a logic quantity Compliment: the inverse of a variable Literal: a

### Introduction to Boolean Algebra

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

### Boolean Analysis of Logic Circuits

Course: B.Sc. Applied Physical Science (Computer Science) Year & Sem.: IInd Year, Sem - IIIrd Subject: Computer Science Paper No.: IX Paper Title: Computer System Architecture Lecture No.: 7 Lecture Title:

### Formal Methods of Software Design, Eric Hehner, segment 1 page 1 out of 5

Formal Methods of Software Design, Eric Hehner, segment 1 page 1 out of 5 [talking head] Formal Methods of Software Engineering means the use of mathematics as an aid to writing programs. Before we can

### 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

### ENEL 353: Digital Circuits Midterm Examination

NAME: SECTION: L01: Norm Bartley, ST 143 L02: Steve Norman, ST 145 When you start the test, please repeat your name and section, and add your U of C ID number at the bottom of the last page. Instructions:

### 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

### CHAPTER-2 STRUCTURE OF BOOLEAN FUNCTION USING GATES, K-Map and Quine-McCluskey

CHAPTER-2 STRUCTURE OF BOOLEAN FUNCTION USING GATES, K-Map and Quine-McCluskey 2. Introduction Logic gates are connected together to produce a specified output for certain specified combinations of input

### 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

### 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

### Digital Logic Lecture 7 Gate Level Minimization

Digital Logic Lecture 7 Gate Level Minimization By Ghada Al-Mashaqbeh The Hashemite University Computer Engineering Department Outline Introduction. K-map principles. Simplification using K-maps. Don t-care

### 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

### Logic and Computer Design Fundamentals. Chapter 2 Combinational Logic Circuits. Part 3 Additional Gates and Circuits

Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 3 Additional Gates and Circuits Charles Kime & Thomas Kaminski 28 Pearson Education, Inc. (Hyperlinks are active in View

### 6. Combinational Circuits. Building Blocks. Digital Circuits. Wires. Q. What is a digital system? A. Digital: signals are 0 or 1.

Digital Circuits 6 Combinational Circuits Q What is a digital system? A Digital: signals are or analog: signals vary continuously Q Why digital systems? A Accurate, reliable, fast, cheap Basic abstractions

### EE292: Fundamentals of ECE

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

### DIGITAL ARITHMETIC: OPERATIONS AND CIRCUITS

C H A P T E R 6 DIGITAL ARITHMETIC: OPERATIONS AND CIRCUITS OUTLINE 6- Binary Addition 6-2 Representing Signed Numbers 6-3 Addition in the 2 s- Complement System 6-4 Subtraction in the 2 s- Complement

### 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 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.

### This podcast will demonstrate a logical approach as to how a computer adds through logical gates.

This podcast will demonstrate a logical approach as to how a computer adds through logical gates. A computer is a programmable machine that receives input, stores and manipulates data, and provides output

### 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

### Propositional Calculus. Math Foundations of Computer Science

Propositional Calculus Math Foundations of Computer Science Propositional Calculus Objective: To provide students with the concepts and techniques from propositional calculus so that they can use it to

### EECS 140/141 Introduction to Digital Logic Design Fall Semester 2016 Exam #1 Date: 3 October 2016

EECS 4/4 Introduction to Digital Logic Design Fall Semester 26 Exam # Date: 3 October 26 NAME: KUID: General Instructions. This exam is closed-book. You are allowed a non-communicating calculator and one

### EEE130 Digital Electronics I Lecture #4_1

EEE130 Digital Electronics I Lecture #4_1 - Boolean Algebra and Logic Simplification - By Dr. Shahrel A. Suandi 4-6 Standard Forms of Boolean Expressions There are two standard forms: Sum-of-products form

### 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

### SYNERGY INSTITUTE OF ENGINEERING & TECHNOLOGY,DHENKANAL LECTURE NOTES ON DIGITAL ELECTRONICS CIRCUIT(SUBJECT CODE:PCEC4202)

Lecture No:5 Boolean Expressions and Definitions Boolean Algebra Boolean Algebra is used to analyze and simplify the digital (logic) circuits. It uses only the binary numbers i.e. 0 and 1. It is also called

### 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-

### 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

### 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

### DIGITAL SYSTEM DESIGN

DIGITAL SYSTEM DESIGN UNIT I: Introduction to Number Systems and Boolean Algebra Digital and Analog Basic Concepts, Some history of Digital Systems-Introduction to number systems, Binary numbers, Number

### Bits, Words, and Integers

Computer Science 52 Bits, Words, and Integers Spring Semester, 2017 In this document, we look at how bits are organized into meaningful data. In particular, we will see the details of how integers are

### Chapter 3 Data Representation

Chapter 3 Data Representation The focus of this chapter is the representation of data in a digital computer. We begin with a review of several number systems (decimal, binary, octal, and hexadecimal) and

### 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

### ENGIN 112. Intro to Electrical and Computer Engineering

ENIN 2 Intro to Electrical and Computer Engineering Lecture 6 More Boolean Algebra ENIN2 L6: More Boolean Algebra September 5, 23 A B Overview Epressing Boolean functions Relationships between algebraic

### Combinational Circuits

Combinational Circuits Q. What is a combinational circuit? A. Digital: signals are or. A. No feedback: no loops. analog circuits: signals vary continuously sequential circuits: loops allowed (stay tuned)

### 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

### 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

### 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

### 2. BOOLEAN ALGEBRA 2.1 INTRODUCTION

2. BOOLEAN ALGEBRA 2.1 INTRODUCTION In the previous chapter, we introduced binary numbers and binary arithmetic. As you saw in binary arithmetic and in the handling of floating-point numbers, there is

### Slide Set 5. for ENEL 353 Fall Steve Norman, PhD, PEng. Electrical & Computer Engineering Schulich School of Engineering University of Calgary

Slide Set 5 for ENEL 353 Fall 207 Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary Fall Term, 207 SN s ENEL 353 Fall 207 Slide Set 5 slide

### Activity Boolean Algebra

Activity 2.1.6 Boolean Algebra Introduction Have you ever had an idea that you thought was so unique that when you told someone else about it, you simply could not believe they thought you were wasting

### Logic, Words, and Integers

Computer Science 52 Logic, Words, and Integers 1 Words and Data The basic unit of information in a computer is the bit; it is simply a quantity that takes one of two values, 0 or 1. A sequence of k bits

### 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 Digital Design. Laboratory Manual. Experiment #3. Boolean Algebra Continued

The Islamic University of Gaza Engineering Faculty Department of Computer Engineering Fall 2017 ECOM 2013 Khaleel I. Shaheen Combinational Digital Design Laboratory Manual Experiment #3 Boolean Algebra

### Combinational Logic Circuits

Chapter 3 Combinational Logic Circuits 12 Hours 24 Marks 3.1 Standard representation for logical functions Boolean expressions / logic expressions / logical functions are expressed in terms of logical

### Lecture #21 March 31, 2004 Introduction to Gates and Circuits

Lecture #21 March 31, 2004 Introduction to Gates and Circuits To this point we have looked at computers strictly from the perspective of assembly language programming. While it is possible to go a great

### EECS150 Homework 2 Solutions Fall ) CLD2 problem 2.2. Page 1 of 15

1.) CLD2 problem 2.2 We are allowed to use AND gates, OR gates, and inverters. Note that all of the Boolean expression are already conveniently expressed in terms of AND's, OR's, and inversions. Thus,

### COMP combinational logic 1 Jan. 18, 2016

In lectures 1 and 2, we looked at representations of numbers. For the case of integers, we saw that we could perform addition of two numbers using a binary representation and using the same algorithm that

### 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

### 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

### 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,

### Information Science 1

Information Science Boolean Expressions Week College of Information Science and Engineering Ritsumeikan University Topics covered l Terms and concepts from Week 9 l Binary (Boolean) logic History Boolean

### R10. II B. Tech I Semester, Supplementary Examinations, May

SET - 1 1. a) Convert the following decimal numbers into an equivalent binary numbers. i) 53.625 ii) 4097.188 iii) 167 iv) 0.4475 b) Add the following numbers using 2 s complement method. i) -48 and +31

### 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

### Midterm Exam Review. CS 2420 :: Fall 2016 Molly O'Neil

Midterm Exam Review CS 2420 :: Fall 2016 Molly O'Neil Midterm Exam Thursday, October 20 In class, pencil & paper exam Closed book, closed notes, no cell phones or calculators, clean desk 20% of your final

### 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

### 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

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

### 60-265: Winter ANSWERS Exercise 4 Combinational Circuit Design

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

### 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

### CS61C : Machine Structures

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

### LOGIC CIRCUITS. Kirti P_Didital Design 1

LOGIC CIRCUITS Kirti P_Didital Design 1 Introduction The digital system consists of two types of circuits, namely (i) Combinational circuits and (ii) Sequential circuit A combinational circuit consists

### 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

### Experimental Methods I

Experimental Methods I Computing: Data types and binary representation M.P. Vaughan Learning objectives Understanding data types for digital computers binary representation of different data types: Integers

### Computer Organization and Levels of Abstraction

Computer Organization and Levels of Abstraction Announcements Today: PS 7 Lab 8: Sound Lab tonight bring machines and headphones! PA 7 Tomorrow: Lab 9 Friday: PS8 Today (Short) Floating point review Boolean

### 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

### LECTURE 4. Logic Design

LECTURE 4 Logic Design LOGIC DESIGN The language of the machine is binary that is, sequences of 1 s and 0 s. But why? At the hardware level, computers are streams of signals. These signals only have two

### 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

### 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