CS February 17
|
|
- Griselda Matthews
- 5 years ago
- Views:
Transcription
1 Discrete Mathematics CS 26 February 7
2 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) = xy + xz, and F=G (recall the truth table from an earlier slide) Also, note the distributive property: x(y+z) = xy + xz via the distributive law 2
3 Boolean Functions Two Boolean expressions e and e 2 that represent the exact same function F are called equivalent x x 2 x 3 F(x,x 2,x 3 ) F(x,x 2,x 3 ) = x (x 2 +x 3 )+x x 2 x 3 F(x,x 2,x 3 ) = x x 2 +x x 3 +x x 2 x 3 3
4 Boolean Functions More equivalent Boolean expressions: (x + y)z = xyz + xyz + xyz (x + y)z = xz + yz distributive = xz + yz identity = x(y + y)z + (x + x)yz unit = xyz + xyz + xyz + xyz distributive = xyz + xyz + xyz idempotent We ve expanded the initial expression into its sum of products form. 4
5 Boolean Functions More equivalent Boolean expressions: xy + z =?? = xy + z identity = xy(z + z) + (x + x)z unit = xyz + xyz + xz + xz distributive = xyz + xyz + x(y + y)z + x(y + y)z unit = xyz + xyz + xyz + xyz + xyz + xyz distributive = xyz + xyz + xyz + xyz + xyz idempotent 5
6 Representing Boolean Functions How to construct a Boolean expression that represents a Boolean Function? represents a Boolean Function? z y x F F(x y z) = if and only if: ( )( )( )+( ) + ( ) + F(x, y, z) = if and only if: (-x)(y)(-z) + (-x)yz + x(-y)z + xyz 6
7 Representing Boolean Functions How to construct a Boolean expression that represents a Boolean Function? x y z F F(x, y, z) = x y z + x y z + x y z + x y z z 7
8 Boolean Identities Double complement: x = x Idempotent laws: x + x = x, x x = x Identity laws: x + = x, x = x Domination laws: x + =, x = Commutative laws: x + y = y + x, x y = y x Associative laws: x + (y + z) = (x + y) + z x (y z) = (x y) z Distributive laws: x + y z = (x + y) (x + z) x (y + z) = x y + x z De Morgan s laws: (x y) = x + y, y (x + y) = x y Absorption laws: x + x y = x, x (x + y) = x the Unit Property: x + x = and Zero Property: x x = 8
9 Boolean Identities Absorption law: Show that x (x + y) = x ) x (x + y) = (x + ) (x + y) identity 2) = x + y distributive 3) = x + y commutative 4) = x + domination 5) = x identity 9
10 Dual Expression (related to identity pairs) The dual e d of a Boolean expression e is obtained by exchanging + with, and with in e. e Example: e = xy + zw e = x + y + e d =(x + y)(z + w) e d = x y Duality principle: e e 2 iff e d e 2 d x (x + y) = x iff x + xy = x (absorption)
11 Dual Function The dual of a Boolean function F represented by a Boolean expression is the function represented by the dual of this expression. The dual function of F is denoted by F d The dual function, denoted by F d, does not depend on the particular Boolean expression used to represent F.
12 Recall: Boolean Identities Double complement: x = x Idempotent laws: x + x = x, x x = x Identity laws: x + = x, x = x Domination laws: x + =, x = Commutative laws: x + y = y + x, x y = y x Associative laws: x + (y + z) = (x + y) + z x (y z) = (x y) z Distributive laws: x + y z = (x + y) (x + z) x (y + z) = x y + x z De Morgan s laws: (x y) = x + y, y (x + y) = x y Absorption laws: x + x y = x, x (x + y) = x the Unit Property: x + x = and Zero Property: x x = 2
13 Boolean Expressions Sets Propositions Identity Laws x = x, x = x +, Complement Laws x x =, x x = Associative Laws (x y) z = x (y z) (x y) z = x (y z) Commutative Laws x y = y x, x y = y x Distributive Laws x ( y z) = (x y) (x z) x (y z) = (x y) (x z) Propositional logic has operations,, and elements T and F such that the above properties hold for all x, y, and z. 3
14 DNF: Disjunctive Normal Form A literal is a Boolean variable or its complement. A minterm of Boolean variables x,,xx n is a Boolean product of n literals y y n, where y i is either the literal x i or its complement x. i minterms Example: x y z + x y z + x y z Disjunctive Normal Form: sum of products We have seen how to develop a DNF expression for a function if we re given the function s truth table. 4
15 CNF: Conjunctive Normal Form A literal is a Boolean variable or its complement. A maxterm of Boolean variables x,,xx n is a Boolean sum of n literals y y n, where y i is either the literal x i or its complement x. i maxterms Example: (x +y + z) (x + y + z) (x + y +z) Conjuctive Normal Form: product of sums 5
16 Conjunctive Normal Form To find the CNF representation of a Boolean function F. Find the DNF representation of its complement F 2. Then complement both sides and apply DeMorgan s laws to get F: x y z F F F = 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 F = (x y z) (x y z) (x y z) (x y z) = (x+y+z) (x+y+z) (x+y+z) (x+y+z) How do you build the CNF directly from the table? 6
17 Functional Completeness Since every Boolean function can be expressed in terms of,+,, we say that the set of operators {,+, } is functionally complete. {+, } is functionally complete (no way!) xy = (x+y) {, } is functionally complete (note: x + y = xy) NAND and NOR are also functionally complete, each by itself (as a singleton set). Recall x NAND y is true iff x or y (or both) are false and x NOR y is true iff both x and y are false. 7
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:
More informationPermutation 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
More informationUnit-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
More informationMenu. 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 ):
More informationPropositional Calculus: Boolean Algebra and Simplification. CS 270: Mathematical Foundations of Computer Science Jeremy Johnson
Propositional Calculus: Boolean Algebra and Simplification CS 270: Mathematical Foundations of Computer Science Jeremy Johnson Propositional Calculus Topics Motivation: Simplifying Conditional Expressions
More informationChapter 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
More informationBoolean 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
More informationCircuit 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
More informationBinary 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
More informationPhiladelphia 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
More information3. According to universal addressing, what is the address of vertex d? 4. According to universal addressing, what is the address of vertex f?
1. Prove: A full m-ary tree with i internal vertices contains n = mi + 1 vertices. 2. For a full m-ary tree with n vertices, i internal vertices, and l leaves, prove: (i) i = (n 1)/m and l = [(m 1)n +
More informationBoolean 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
More informationIntroduction to Boolean logic and Logical Gates
Introduction to Boolean logic and Logical Gates Institute of Statistics Fall 2014 We saw the importance of the binary number system for data representation in a computer system. We ll see that the construction
More informationDefinitions. 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
More informationDSAS 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
More informationChapter 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
More information24 Nov Boolean Operations. Boolean Algebra. Boolean Functions and Expressions. Boolean Functions and Expressions
24 Nov 25 Boolean Algebra Boolean algebra provides the operations and the rules for working with the set {, }. These are the rules that underlie electronic circuits, and the methods we will discuss are
More informationVariable, 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
More informationBOOLEAN 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)
More informationNormal Forms Boolean Satisfiability Sectoins 5.3 and 5.5. Prof. Sandy Irani
Normal Forms Boolean Satisfiability Sectoins 5.3 and 5.5 Prof. Sandy Irani Normal Forms for Boolean Expressions Boolean expressions can be put into standardized forms. Useful for manipulating expressions
More informationIntroduction 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
More informationUNIT 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
More informationBoolean 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: +
More information2.1 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
More informationCSC Discrete Math I, Spring Sets
CSC 125 - Discrete Math I, Spring 2017 Sets Sets A set is well-defined, unordered collection of objects The objects in a set are called the elements, or members, of the set A set is said to contain its
More informationTA: Jade Cheng ICS 241 Recitation Lecture Notes #12 November 13, 2009
TA: Jade Cheng ICS 241 Recitation Lecture Notes #12 November 13, 2009 Recitation #12 Question: Use Prim s algorithm to find a minimum spanning tree for the given weighted graph. Step 1. Start from the
More informationboolean.py Documentation
boolean.py Documentation Release 3.2 Sebastian Krämer Apr 13, 2018 Contents 1 User Guide 1 1.1 Introduction............................................... 1 1.2 Installation................................................
More informationCS8803: 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
More informationGate-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
More informationUNIT-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?
More informationComputer 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
More informationLogic Design: Part 2
Orange Coast College Business Division Computer Science Department CS 6- Computer Architecture Logic Design: Part 2 Where are we? Number systems Decimal Binary (and related Octal and Hexadecimal) Binary
More informationCombinational 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
More informationX 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
More informationLecture 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
More informationGate 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
More informationComputer 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
More informationIntroduction 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
More informationIntroduction 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
More informationGate 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 =
More information2008 The McGraw-Hill Companies, Inc. All rights reserved.
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
More informationExperiment 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
More informationBoolean 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
More information1. 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
More informationELCT201: 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
More informationDigital 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
More informationTo prove something about all Boolean expressions, we will need the following induction principle: Axiom 7.1 (Induction over Boolean expressions):
CS 70 Discrete Mathematics for CS Spring 2005 Clancy/Wagner Notes 7 This lecture returns to the topic of propositional logic. Whereas in Lecture Notes 1 we studied this topic as a way of understanding
More informationSpring 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
More informationLecture 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
More informationCode 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
More information01 Introduction to Digital Logic. ENGR 3410 Computer Architecture Mark L. Chang Fall 2008
Introduction to Digital Logic ENGR 34 Computer Architecture Mark L. Chang Fall 28 Acknowledgements Patterson & Hennessy: Book & Lecture Notes Patterson s 997 course notes (U.C. Berkeley CS 52, 997) Tom
More informationGate-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
More informationPropositional Calculus. CS 270: Mathematical Foundations of Computer Science Jeremy Johnson
Propositional Calculus CS 270: Mathematical Foundations of Computer Science Jeremy Johnson Propositional Calculus Objective: To provide students with the concepts and techniques from propositional calculus
More informationChapter 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
More informationCode 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,
More information[Ch 6] Set Theory. 1. Basic Concepts and Definitions. 400 lecture note #4. 1) Basics
400 lecture note #4 [Ch 6] Set Theory 1. Basic Concepts and Definitions 1) Basics Element: ; A is a set consisting of elements x which is in a/another set S such that P(x) is true. Empty set: notated {
More informationChapter 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
More informationDigital Design. Chapter 4. Principles Of. Simplification of Boolean Functions
Principles Of Digital Design Chapter 4 Simplification of Boolean Functions Karnaugh Maps Don t Care Conditions Technology Mapping Optimization, Conversions, Decomposing, Retiming Boolean Cubes for n =,
More informationBawar 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
More informationExperiment 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.
More informationBoolean Functions (10.1) Representing Boolean Functions (10.2) Logic Gates (10.3)
Chapter (Part ): Boolean Algebra Boolean Functions (.) Representing Boolean Functions (.2) Logic Gates (.3) It has started from the book titled The laws of thought written b George Boole in 854 Claude
More informationBawar 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
More informationTo prove something about all Boolean expressions, we will need the following induction principle: Axiom 7.1 (Induction over Boolean expressions):
CS 70 Discrete Mathematics for CS Fall 2003 Wagner Lecture 7 This lecture returns to the topic of propositional logic. Whereas in Lecture 1 we studied this topic as a way of understanding proper reasoning
More informationQUESTION 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
More informationReview. 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
More informationIT 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.
More informationBOOLEAN 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
More informationSimplification 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.
More information01 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
More informationGate-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
More informationReview: Standard forms of expressions
Karnaugh maps Last time we saw applications of Boolean logic to circuit design. The basic Boolean operations are AND, OR and NOT. These operations can be combined to form complex expressions, which can
More informationAssignment (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
More informationObjectives: 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-
More informationMathematical Logic Prof. Arindama Singh Department of Mathematics Indian Institute of Technology, Madras. Lecture - 9 Normal Forms
Mathematical Logic Prof. Arindama Singh Department of Mathematics Indian Institute of Technology, Madras Lecture - 9 Normal Forms In the last class we have seen some consequences and some equivalences,
More informationMixed Integer Linear Programming
Mixed Integer Linear Programming Part I Prof. Davide M. Raimondo A linear program.. A linear program.. A linear program.. Does not take into account possible fixed costs related to the acquisition of new
More information2.2 Set Operations. Introduction DEFINITION 1. EXAMPLE 1 The union of the sets {1, 3, 5} and {1, 2, 3} is the set {1, 2, 3, 5}; that is, EXAMPLE 2
2.2 Set Operations 127 2.2 Set Operations Introduction Two, or more, sets can be combined in many different ways. For instance, starting with the set of mathematics majors at your school and the set of
More informationCombinatorial Algorithms. Unate Covering Binate Covering Graph Coloring Maximum Clique
Combinatorial Algorithms Unate Covering Binate Covering Graph Coloring Maximum Clique Example As an Example, let s consider the formula: F(x,y,z) = x y z + x yz + x yz + xyz + xy z The complete sum of
More informationChapter 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
More informationLecture (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
More informationELCT201: 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
More information2.6 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
More informationAnnouncements. 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
More informationCombinational 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
More informationCS470: Computer Architecture. AMD Quad Core
CS470: Computer Architecture Yashwant K. Malaiya, Professor malaiya@cs.colostate.edu AMD Quad Core 1 Architecture Layers Building blocks Gates, flip-flops Functional bocks: Combinational, Sequential Instruction
More informationComputer Engineering Chapter 3 Boolean Algebra
Computer Engineering Chapter 3 Boolean Algebra Hiroaki Kobayashi 5/30/2011 Ver. 06102011 5/30/2011 Computer Engineering 1 Agenda in Chapter 3 What is Boolean Algebra Basic Boolean/Logical Operations (Operators)
More informationComputer Organization and Levels of Abstraction
Computer Organization and Levels of Abstraction Announcements PS8 Due today PS9 Due July 22 Sound Lab tonight bring machines and headphones! Binary Search Today Review of binary floating point notation
More informationLECTURE 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
More informationR.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
More informationCode 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
More informationHomework 1. Due Date: Wednesday 11/26/07 - at the beginning of the lecture
Homework 1 Due Date: Wednesday 11/26/07 - at the beginning of the lecture Problems marked with a [*] are a littlebit harder and count as extra credit. Note 1. For any of the given problems make sure that
More information(a) (4 pts) Prove that if a and b are rational, then ab is rational. Since a and b are rational they can be written as the ratio of integers a 1
CS 70 Discrete Mathematics for CS Fall 2000 Wagner MT1 Sol Solutions to Midterm 1 1. (16 pts.) Theorems and proofs (a) (4 pts) Prove that if a and b are rational, then ab is rational. Since a and b are
More informationENGINEERS 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
More information2. 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
More informationInformation 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
More informationSIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road QUESTION BANK (DESCRIPTIVE)
SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 517583 QUESTION BANK (DESCRIPTIVE) Subject with Code : STLD(16EC402) Year & Sem: II-B.Tech & I-Sem Course & Branch: B.Tech
More information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,
More information9/19/12. Why Study Discrete Math? What is discrete? Sets (Rosen, Chapter 2) can be described by discrete math TOPICS
What is discrete? Sets (Rosen, Chapter 2) TOPICS Discrete math Set Definition Set Operations Tuples Consisting of distinct or unconnected elements, not continuous (calculus) Helps us in Computer Science
More informationChapter 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
More informationS1 Teknik Telekomunikasi Fakultas Teknik Elektro FEH2H3 2016/2017
S1 Teknik Telekomunikasi Fakultas Teknik Elektro FEH2H3 2016/2017 Karnaugh Map Karnaugh maps Last time we saw applications of Boolean logic to circuit design. The basic Boolean operations are AND, OR and
More informationIdentity. p ^ T p p _ F p. Idempotency. p _ p p p ^ p p. Associativity. (p _ q) _ r p _ (q _ r) (p ^ q) ^ r p ^ (q ^ r) Absorption
dam lank pring 27 E 3 Identity Pre-Lecture Problem Do it! Do it now! What are you waiting for?! Use Logical Equivalences to show!! $! & & $! p ^ T p p _ F p Domination p _ T T p ^ F F Idempotency p _ p
More information