# Chapter 2 Boolean algebra and Logic Gates

Size: px
Start display at page:

Download "Chapter 2 Boolean algebra and Logic Gates"

Transcription

1 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 and solve for unknowns. Originally, Boolean algebra which was formulated by George Boole, an English mathematician (85-864) described propositions whose outcome would be either true or false. In computer work it is used in addition to describe circuits whose state can be either (true) or (false).using the relations defined in the AND, OR and NOT operation, a number of postulates are stated. In this chapter, we are going to learn What is called as Boolean algebra? Some basic theorems of Boolean algebra Basic logic gates used in logic circuits In what way Boolean algebra used in building the logic circuits 2.2 Definition of Boolean algebra In 854 George Boole introduced a systematic treatment of logic and developed for thispurpose an algebraic system now called Boolean algebra. Boolean algebra is an algebraic structure defined on a set of elements B togetherwith two binary operators + and.(dot) provided the following postulates aresatisfied: Let us assume that A and B are two logical variables, which can either be true or false. The + operator is the union or the OR operator. The is the intersection or the AND operator. The following are some of the laws of Boolean Algebra:. A+B=B+A 2. A B=B A 3. (A+B)+C = A+(B+C) 4. (A.B).C = (A.B).C 5. A+(B.C) = (A+B).(A+C) 6. A.(B+C) = (A.B)+(A.C) 7. A+=A 8. A+= 9. A+A=A. A.A=A. A+A = 2. A.A =

2 Where A is complement of A (Or A ). Venn diagrams can be used to illustrate the above. Figure Basic theorems of Boolean algebra From the postulates defined above we can prove some of the Boolean theorems which are often used in simplification of Boolean functions. Simplification of Boolean functions will reduce the number of gates required to build the logic circuit and thereby reduces the cost of the circuit. So we have to be familiar with the postulates and theorems. Duality theorem: It states that every algebraic expression remains valid if the operator and identity elements are interchanged. For example x+ = x is dual of x. = x Theorem (a): x+x = x Proof: x+x = (x+x). = (x+x).(x+x ) =x+x.x =x+ = x

3 Theorem (b): x.x = x Proof: x.x = (x.x)+ = (x.x)+(x.x ) =x.(x+x ) =x. = x Note that theorem (b) is dual of (a). Theorem 2(a): x+ = Proof: x+ = (x+). = (x+).(x+x ) =x+.x = x+x = Theorem 2(b): x. = (by duality principle) Theorem 3: (x ) = x Theorem 4(a): x+(y+z) = (x+y)+z Theorem 4(b): x.(y.z) = (x.y).z Theorem 5(a)-DeMorgan s theorem: (x+y) = x.y Theorem 5(b)-DeMorgan s theorem: (x.y) = x +y Theorem 6(a) absorption theorem: x+xy = x Proof: x+xy = x.+x.y =x.(+y) =x. =x Theorem 6(b) absorption theorem: x.(x+y) = x Proof: x.(x+y) = (x+).(x+y) =x+(.y) =x+ =x

4 2.4 Logic gates Basic logic gates All digital systems can be constructed by only three basic logic gates. These basic gates are called the NOT gate, AND gate and the OR gate. NAND and NOR are called as universal gates as using only NAND gates (or only NOR gates) we can construct the logic circuits. Ex-OR and Ex-NOR are two exclusive gates used in the logic circuits. NOT gate The NOT gate is a circuit which produces at its output the negated (inverted) version of its input logic. The circuit is also known as an inverter. If the input variable is A, the inverted output is written as A or A. AND gate Figure 2.2 The AND gate is a circuit which gives a high output (logic ) if all its inputs are high. A dot is used to indicate the AND operation. Figure 2.3

5 OR gate The OR gate is a circuit which gives a high output if one or more of its inputs are high. A plus sign (+) is used to indicate the OR operation. NAND gate Figure 2.4 The NAND gate is a NOT-AND circuit which is equivalent to an AND circuit followed by a NOT circuit. The output of the NAND gate is high if any of its inputs is low. NOR gate Figure 2.5 The NOR gate is a NOT-OR circuit which is equivalent to an OR circuit followed by a NOT circuit. The output of the NOR gate is low if any of its inputs is high.

6 Figure 2.6 Ex-OR gate The Exclusive-OR gate is a circuit which gives a high output if either of its two inputs is high, but not both. A encircled plus sign is used to indicate the Ex-OR operation. Figure Boolean functions and Expressions Boolean functions are formed by using binary variables. Binary variable means that a variable could take either or. So the output of Boolean function is also either or. Boolean expressions are generally consists of NOT, AND, OR operations. A Boolean expression can be implemented in practice by using the logic gates. General precedence of order is followed in evaluating the Boolean expression, i.e., Parenthesis, NOT, AND, OR. Let us consider a Boolean expression F = x.y +z. The variables in the expression are F, x, y and z. In this Boolean expression x, y and z are input variables and F is the variable used to specify the output. Now we are going to draw truth table for this expression.

7 Table 2. x y z y x.y F = x.y +z Three input variables x, y and z are written by possible combinations. As each variable can take either or, three variables can take 2 3 = 8 combinations starting from,,.. By using order of precedence rule, first calculate the y (NOT first), then x.y (AND next) and finally calculate (x.y )+z (OR function). Truth table is the table which consists of possible inputs and their corresponding output. So for Boolean expression F = x.y +z, the truth table is given as Table 2.2 Input Output x y z F = x.y +z

8 Note that this table contains only inputs (x, y and z) and output (F). Let us consider another Boolean expression F2 = (x+y ). We will construct truth table for this Boolean expression now. By order of precedence rule, first we have to compute (x+y ), then take the complement of (x+y ) to get output F. Table 2.3 x y y x+y F2 = (x+y ) So the truth table for the Boolean expression F2 = (x+y ) is given as Table 2.4 Input Output x y F2 So now we are able to construct truth table for given Boolean expression. Note that many Boolean expressions can give the same output. Two Boolean expressions that give the same output is said to be equivalent. For example consider the Boolean expression, F3 = x.y

9 Table 2.5 x y X F3 = x.y The truth table for the expression F3 = x.y is given as Table 2.6 Input Output x y F3 By comparing the truth table of F2 and F3, both Boolean expressions give the same output. So we can say that F2 = (x+y ) and F3 = x.y are equivalent. 2.6 Canonical and standard terms To conveniently specify the truth table in Boolean algebra, canonical and standard terms are used. As we have just discussed many Boolean expressions may give the same output. But while using canonical form of Boolean expression there is only one canonical form of Boolean expression to specify one truth table. To map the truth table into canonical Boolean expression, we have two standard forms. (i) the sum of minterms (product terms) (ii) the product of maxterms (sum terms). Let us know what is meant by minterms and maxterms.

10 Minterms Consider three input variables x, y and z. As binary variables can take either or, there could be eight possible combinations (2 3 = 8) i.e.,,,.. A zero is considered as complemented variable, one is considered as true variable. Thus 3 input variables have 8 minterms. x y z, x y z, x yz, x yz, xy z, xy z, xyz, xyz Minterms are also called as product terms as x,y and z are ANDed either in complemented or true form. Maxterms Note that here zero is considered as true variable, one is considered as complemented variable. Thus 3 input variables have 8 minterms. x+y+z, x+y+z, x+y +z, x+y +z, x +y+z, x +y+z, x +y +z, x +y +z Maxterms are also called as sum terms as x,y and z are ORed either in complemented or true form. Table 2.7 Input x Minterms Maxterms y z Term Designation Term Designation x y z m x+y+z M x y z m x+y+z M x yz m2 x+y +z M2 x yz m3 x+y +z M3 xy z m4 x +y+z M4 xy z m5 x +y+z M5 xyz m6 x +y +z M6 xyz m7 x +y +z M7 The above table shows minterms and maxterms for three input variables. In the same way two input variables will have 4 minterms and 4 maxterms. Four input variables will have 6 minterms and maxterms. So we know the minterms and maxterms now. Let us see how to map a truth table into Boolean expression using these minterms and maxterms. First consider the truth table with two input variables say A and B and an output variable X.

11 Table 2.8 Input A Output B X To specify this truth table in canonical form of Boolean expression using minterms, consider the output X wherever which is equal to one. The output is whenever A= and B= or A= and B=. If A= and B=, the corresponding minterm is A B. If A= and B=, the corresponding minterm is AB. So the Boolean expression is X = A B+AB This expression can also be written as X = m+m2 or X = (m,m2) or simply X= (,2). This is very convenient way of representing the truth table and it is unique. As output is expressed as sum of minterms, this type of expression is called as sum of product expression or SOP. The same truth table can be expressed using maxterms also. For maxterms we should consider the output X wherever it is equal to zero. The output X is zero if the input A= and B= or A= and B=. If A= and B=, the corresponding maxterm is (A+B). If A= and B=, the corresponding maxterms is (A +B ). So the Boolean expression is X = (A+B)(A +B ) This expression can also be written as X = M, M3 or X = (M,M3) or simply X= (,3). As output is expressed as product of maxterms, this type of expression is called as product of sum expression or POS. How to convert the Boolean algebra into Canonical forms There may be a question raised in your mind that how to convert any Boolean expression into canonical forms. It is very simple to convert any Boolean algebra into its equivalent canonical forms. The need for conversion is as already said it is convenient way to represent in canonical form and it is unique. Let us consider a Boolean expression F = A+B C. We need to convert this Boolean expression into its equivalent sum of product expression (SOP). By seeing the expression we can easily say that it has three inputs A, B and C and one output F. we will build a truth table first.

12 Table 2.9 A B C B B C F=A+B C From the table we can easily write F = A B C+AB C +AB C+ABC +ABC or F = m+m4+m5+m6+m7 or F = Σ(, 4, 5, 6, 7). Let consider another Boolean expression F = xy+x z and convert it into POS expression. Again draw the truth table Table 2. x y z X x z xy F=xy+x z From the table we can easily write F = (x+y+z)(x+y +z)(x +y+z)(x +y+z ) or F= M.M2.M4.M5 or F = Π(, 2, 4, 5).

13 How to convert SOP expression into POS and vice versa To convert SOP expression into POS and vice versa, writes the missing terms because POS and SOP are complement of each other. Consider the previous example F = xy+x z, and we have found that its POS expression is that F = Π(, 2, 4, 5). To convert this expression into its equivalent SOP form, find the missing terms. Missing terms are here, 3, 6, 7. So we can write SOP expression as F = Σ(, 3, 6, 7). 2.7 Implementation of Boolean function using logic gates Implementation of Boolean function using basic logic gates It is possible to implement any Boolean expression, using logic gates in practical. As a first case, we will construct the logic circuit for some Boolean expression using only basic gates like NOT, AND, OR gates. Later we will learn to implement the Boolean function using only NAND gates or using only NOR gates. Let consider a Boolean function F = x+y z. What are all the logic operations in this Boolean function? As order of precedence first NOT operation, then AND operation and OR operation. So we need a NOT gate, AND gate and OR gate to implement this function. Figure 2.8 First obtain the required complemented variable using NOT gate(s). In this particular example, we need complement of y only. Then using AND and OR gates we can complete this circuit.

14 Figure 2.9 Consider another Boolean function F2 = x y z+x yz+xy. As we have just seen, first obtain the required complement function. Note that for x and y only complemented variable needed, not for z. Figure 2.

15 Figure 2. Implementation of Boolean function using universal logic gates NAND implementation NAND and NOR gates are easier to fabricate and are the basic gates used in all IC digital logic families. So implementing Boolean function using NAND and NOR gates are necessary. First we implement basic logic gates NOT, AND and OR using only NAND gates. A one input NAND gate behaves like an inverter or NOT gate. Figure 2.2 As NAND function is complemented function of AND gate, to derive AND gate using NAND gates again it should be complemented. So connecting a NOT gate at the output of NAND gate will provide AND function. Figure 2.3 NOT gate in the output of NAND gate can be replaced by single input NAND gate as single input NAND gate will act as NOT gate or inverter.

16 Figure 2.4 OR gate can be implemeted by using NAND gates only by using DeMorgan s theorem. As per DeMorgan s theorem, (x+y) = x.y. So x+y = (x.y ). So first we need x and y which can be obtained by connecting a single input NAND gate to each input. Figure 2.5 Then x and y are connected to a two input NAND gate to obtain (x.y ) which is equal to OR function x+y. Figure 2.6 Sometimes this circuit is simply represented by using a bubble for inverter operation as shown below. Figure 2.7 To implement a Boolean function uisng only NAND gates Implement the Boolean function using basic logic gates i.e. NOT, AND, OR. Replace all NOT gates by single input NAND gates. Replace all the AND gates by NAND gates adding a bubble at the output of each AND gate. Replace all the OR gates by NAND gates adding a bubble at the inputs of each OR gate. If two bubbles are in the same line, then they will be get cancelled out naturally. If only one bubble is there in a line, then to compensate the invert operation add one more single input NAND gate in that line.

17 Example Consider a Boolean function F = AB+CD. We implement this Boolean function using only NAND gates. Let us draw the logic circuit using basic logic gates. Figure 2.8 Replace AND by NAND putting a bubble at the output of AND gate and replace OR gate by putting bubbles at the inputs of OR gate. Figure 2.9 As two bubbles are in the same lines, two invert functions will cancel each other. So we can replace all the gates by NAND gates. Figure 2.2

18 Example Consider another Boolean fucntion F = A(B+CD)+BC. We implement this Boolean function now using only NAND gates. Let us draw the logic circuit using basic logic gates first. Figure 2.2 First replace the NOT gate by single input NAND gate. Figure 2.22

19 Replace AND by NAND putting a bubble at the output of AND gate and replace OR gate by putting bubbles at the inputs of OR gate. Figure 2.23 As one bubble is there in two lines we have to add inverter to compensate it. Figure 2.24 Now all the gates can be replaced as NAND gates.

20 Figure 2.25 NOR implementation We implement basic logic gates NOT, AND and OR using only NOR gates. A one input NOR gate behaves like an inverter or NOT gate. Figure 2.26 As NOR function is complemented function of OR gate, to derive OR gate using NOR gates again it should be complemented. So connecting a NOT gate at the output of NOR gate will provide OR function. Figure 2.27 NOT gate in the output of NOR gate can be replaced by single input NOR gate as single input NOR gate will act as NOT gate or inverter.

21 Figure 2.28 AND gate can be implemeted by using NOR gates only by using DeMorgan s theorem. As per DeMorgan s theorem, (x.y) = x +y. So x.y = (x +y ). So first we need x and y which can be obtained by connecting a single input NOR gate to each input. Figure 2.29 Then x and y are connected to a two input NOR gate to obtain (x +y ) which is equal to AND function x.y. Figure 2.3 Sometimes this circuit is simply represented by using a bubble for inverter operation as shown below. Figure 2.3 To implement a Boolean function uisng only NOR gates Implement the Boolean function using basic logic gates i.e. NOT, AND, OR. Replace all NOT gates by single input NOR gates. Replace all the OR gates by NOR gates adding a bubble at the output of each OR gate. Replace all the AND gates by NOR gates adding a bubble at the inputs of each AND gate. If two bubbles are in the same line, then they will be get cancelled out naturally. If only one bubble is there in a line, then to compensate the invert operation add one more single input NOR gate in that line.

22 Example Consider a Boolean fucntion F = (A+B)(C+D)E. We implement this Boolean function using only NOR gates. Let us draw the logic circuit using basic logic gates first. Figure 2.32 Replace OR by NOR putting a bubble at the output of OR gate and replace AND gate by putting bubbles at the inputs of AND gate. Figure 2.33 As one bubble is there in one line we have to add inverter to compensate it. Figure 2.34

23 Now all the gates can be replaced as NOR gates. 2.8 Summary Figure 2.35 In this chapter we have defined the Boolean algebra by using some postulates. Boolean algebra postulates can be easily proved by using Venn diagram. Then some basic theorems like duality principle, DeMorgan s theorem and absorption theorem have been proved. These theorems are very much useful in the subsequent topics. NOT, AND and OR which are called as basic logic gates and NAND and NOR which are called as universal gates have been given along with their truth table, Boolean functions. Using these gates only we have to implement the digital system throughout this subject. Therefore learning these gates undoubtedly is important to further proceed. Boolean functions are implemented using logic gates. Canonical forms are unique way to represent a digital system and it is most convenient way also to represent the truth table. Finally we have learnt to construct the logic circuit using basic logic gates. Using only NAND or only NOR gates also we can build the logic circuits that is why NAND and NOR are called as universal gates. Review Questions. Demonstrate by means of truth tables the validity of the following identities: (a) DeMorgan's theorem for three variables: (xyz)' = x' + y' + z'. (b) The consensus theorem: xy + x z + yz = xy + x z. 2. Obtain the truth table of the following functions and express each function in sum of minterms and product of maxterms: (a) (xy + z)(y + xz) (b) (A' + B)(B' + C) (c) y'z + wxy' + wxz' + w'x'z 3. For the Boolean function F given in the truth table, express F in sum of minterms in algebraic form.

24 Table 2. x y z F 4. Express the following functions in sum of minterms and product of maxterms: (a) F(A,B, C,D) = B'D + A'D + BD (b) F(x, y, z) = (xy + z)(xz + y) 5. Convert the following to the other canonical form: (a) F(x, y, z) = Σ(, 3, 7) (b) F(A, B, C, D) = Π(,,2,3,4,6,2) 6. Draw the logic diagram corresponding to the following Boolean expressions (a) BC' + AB + ACD (b) (A + B)(C + D)(A' + B + D) (c) (AB + A'B')(CD' + CD) 7. Draw a NAND logic diagram that implements the complement of the following function: F(A, B, C, D)=Σ(,,2,3,4,8,9,2) 8. Draw a logic diagram using only two-input NAND gates to implement the following expression: (AB + A'B')(CD' + C'D)

### 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 ):

### 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

### 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

### 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

### 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.

### 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

### 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

### 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)

### 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

### 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

### 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?

### Chapter 3 Simplification of Boolean functions

3.1 Introduction Chapter 3 Simplification of Boolean functions In this chapter, we are going to discuss several methods for simplifying the Boolean function. What is the need for simplifying the Boolean

### 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

### 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

### 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

### 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 Map Method

Gate Level Minimization Map Method Complexity of hardware implementation is directly related to the complexity of the algebraic expression Truth table representation of a function is unique Algebraically

### Introduction to Computer Architecture

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

### 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

### Boolean Algebra. BME208 Logic Circuits Yalçın İŞLER

Boolean Algebra BME28 Logic Circuits Yalçın İŞLER islerya@yahoo.com http://me.islerya.com 5 Boolean Algebra /2 A set of elements B There exist at least two elements x, y B s. t. x y Binary operators: +

### 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

### 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

### 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

### 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

### 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

### 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

### 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

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)

### 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

### 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

### 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.

### 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

### 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

### DKT 122/3 DIGITAL SYSTEM 1

Company LOGO DKT 122/3 DIGITAL SYSTEM 1 BOOLEAN ALGEBRA (PART 2) Boolean Algebra Contents Boolean Operations & Expression Laws & Rules of Boolean algebra DeMorgan s Theorems Boolean analysis of logic circuits

### 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

### 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

### 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,

### UNIT II. Circuit minimization

UNIT II Circuit minimization The complexity of the digital logic gates that implement a Boolean function is directly related to the complexity of the algebraic expression from which the function is implemented.

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

### 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

### 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

### 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

### 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:

### 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

### 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

### 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. 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

### 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

### 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

### 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

### 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-

### 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

### 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)

### 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

### 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

### 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 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. BOOLEAN ALGEBRA 2.1 INTRODUCTION

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

### Module -7. Karnaugh Maps

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

### 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.

### 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

### Code No: R Set No. 1

Code No: R059210504 Set No. 1 II B.Tech I Semester Regular Examinations, November 2007 DIGITAL LOGIC DESIGN ( Common to Computer Science & Engineering, Information Technology and Computer Science & Systems

### 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

### 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

### Literal Cost F = BD + A B C + A C D F = BD + A B C + A BD + AB C F = (A + B)(A + D)(B + C + D )( B + C + D) L = 10

Circuit Optimization Goal: To obtain the simplest implementation for a given function Optimization is a more formal approach to simplification that is performed using a specific procedure or algorithm

### 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

### 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

### 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

### 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,

### 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

### 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

### Ch. 5 : Boolean Algebra &

Ch. 5 : Boolean Algebra & Reduction elektronik@fisika.ui.ac.id Objectives Should able to: Write Boolean equations for combinational logic applications. Utilize Boolean algebra laws and rules for simplifying

### 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

### To write Boolean functions in their standard Min and Max terms format. To simplify Boolean expressions using Karnaugh Map.

3.1 Objectives To write Boolean functions in their standard Min and Max terms format. To simplify Boolean expressions using. 3.2 Sum of Products & Product of Sums Any Boolean expression can be simplified

### 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

### 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

### 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

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

### 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

### 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

### 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

### 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:

### Logic Gates and Boolean Algebra ENT263

Logic Gates and Boolean Algebra ENT263 Logic Gates and Boolean Algebra Now that we understand the concept of binary numbers, we will study ways of describing how systems using binary logic levels make

### 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

### 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

### 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

### 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)

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

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

### 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

### 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

### CMPE223/CMSE222 Digital Logic

CMPE223/CMSE222 Digital Logic Optimized Implementation of Logic Functions: Strategy for Minimization, Minimum Product-of-Sums Forms, Incompletely Specified Functions Terminology For a given term, each

### Department of Electrical Engineering McGill University ECSE 221 Introduction to Computer Engineering Assignment 2 Combinational Logic

Department of Electrical Engineering McGill University ECSE 221 Introduction to Computer Engineering Assignment 2 Combinational Logic Question 1: Due October 19 th, 2009 A convenient shorthand for specifying

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

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

### Summary. Boolean Addition

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

### MODULE 5 - COMBINATIONAL LOGIC

Introduction to Digital Electronics Module 5: Combinational Logic 1 MODULE 5 - COMBINATIONAL LOGIC OVERVIEW: For any given combination of input binary bits or variables, the logic will have a specific

### 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

### NH 67, Karur Trichy Highways, Puliyur C.F, Karur District DEPARTMENT OF INFORMATION TECHNOLOGY CS 2202 DIGITAL PRINCIPLES AND SYSTEM DESIGN

NH 67, Karur Trichy Highways, Puliyur C.F, 639 114 Karur District DEPARTMENT OF INFORMATION TECHNOLOGY CS 2202 DIGITAL PRINCIPLES AND SYSTEM DESIGN UNIT 1 BOOLEAN ALGEBRA AND LOGIC GATES Review of binary

### 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