CSE 562 Database Systems
|
|
- Anabel Gregory
- 6 years ago
- Views:
Transcription
1 Outline Query Optimization CSE 562 Database Systems Query Processing: Algebraic Optimization Some slides are based or modified from originals by Database Systems: The Complete Book, Pearson Prentice Hall 2 nd Edition 2008 Garcia-Molina, Ullman, and Widom Overview elational algebra level Algebraic Transformations Detailed query plan level Estimate Costs Estimating size of results Estimating # of IOs Generate and compare plans UB CSE 562 UB CSE elational Algebra Optimization Algebraic ewritings: Commutative and Associative Laws Transformation rules (preserve equivalence) What are good transformations? Cartesian Product Commutative S S Associative T S T S Natural Join S S Question 1: Do the above hold for both sets and bags? Question 2: Do commutative and associative laws hold for arbitrary Theta Joins? S T S T UB CSE UB CSE
2 Algebraic ewritings: Commutative and Associative Laws Algebraic ewritings for Selection: Decomposition of Logical Connectives Commutative Associative Union S S S T T S Intersection S S S T T S Does it apply to bags? Question 1: Do the above hold for both sets and bags? Question 2: Is difference commutative and associative? UB CSE UB CSE Algebraic ewritings for Selection: Decomposition of Negation Pushing Selection Through Binary Operators: Union and Difference Question Complete Union Difference Exercise: Do the rules for intersection UB CSE UB CSE
3 Pushing Selection Through Cartesian Product and Join ules: π + σ combined The right direction requires that cond refers to S attributes only Let X = subset of attributes Z = attributes in predicate P (subset of attributes) The right direction requires that cond refers to S attributes only The right direction requires that all the attributes used by cond appear in both and S Exercise: Do the rule for theta join UB CSE UB CSE Pushing Simple Projections Through Binary Operators: Union Pushing Simple Projections Through Binary Operators: Join and Product A projection is simple if it only consists of an attribute list Union B is the list of attributes that appear in A Similar for C Question 1: Does the above hold for both bags and sets? Question 2: Can projection be pushed below intersection and difference? Answer for both bags and sets UB CSE Question: What is B and C? Exercise: Write the rewriting rule that pushes projection below theta join UB CSE
4 ules: π + σ + combined Projection Decomposition Let X = set of attributes Y = set of attributes XY = X U Y Z = Z U {attributes used in cond} UB CSE UB CSE Projection Decomposition Some ewriting ules elated to Aggregation: SUM σ cond [SUM GroupbyList;GroupedAttribute esultattribute ()] SUM GroupbyList;GroupedAttribute esultattribute [σ cond ()] if cond involves only the GroupbyList SUM GL;GA A ( S) PLUS A1,A2:A [(SUM GL;GA A1 ) (SUM GL;GA A2 S)] SUM GL2;A1 A2 [SUM GL1;GA A1 ()] SUM GL2:GA A2 () Question: does the above hold for both bags and sets? UB CSE UB CSE
5 Derived ules: σ + combined Derivation for first one More ules can be Derived: σp q ( S) = [σp ()] [σq (S)] σp q m ( S) = σm[σp () σq (S)] σpvq ( S) = [σp () S] U [ σq (S)] σp q ( S) = σp[σq ( S)] = σp[ σq (S)] = [σp ()] [σq (S)] p only at q only at S m at both and S UB CSE UB CSE Which are good transformations? Conventional Wisdom: Do Projects Early σp1 p2 () σp1[σp2 ()] σp ( S) [σp ()] S S S πx[σp ()] πx{σp[πxz ()]} UB CSE UB CSE
6 But More Transformations in Textbook What if we have A, B indexes? B = cat A=3 Eliminate common sub-expressions Other operations: duplicate elimination Intersect pointers to get pointers to matching tuples UB CSE UB CSE Bottom line No transformation is always good at the logical query plan level Usually good: early selections elimination of Cartesian products elimination of redundant sub-expressions Many transformations lead to promising plans Commuting/rearranging joins In practice too combinatorially explosive to be handled as rewriting of logical query plan UB CSE
Databases 2011 The Relational Algebra
Databases 2011 Christian S. Jensen Computer Science, Aarhus University What is an Algebra? An algebra consists of values operators rules Closure: operations yield values Examples integers with +,, sets
More informationInformation Systems for Engineers. Exercise 8. ETH Zurich, Fall Semester Hand-out Due
Information Systems for Engineers Exercise 8 ETH Zurich, Fall Semester 2017 Hand-out 24.11.2017 Due 01.12.2017 1. (Exercise 3.3.1 in [1]) For each of the following relation schemas and sets of FD s, i)
More informationInformation Systems for Engineers. Exercise 5. ETH Zurich, Fall Semester Hand-out Due
Information Systems for Engineers Exercise 5 ETH Zurich, Fall Semester 2017 Hand-out 27.10.2017 Due 03.11.2017 Reading material: Chapter 2.4 in [1]. Lecture slides 4. 1. Given the two tables below, write
More informationRelational Algebra on Bags. Why Bags? Operations on Bags. Example: Bag Selection. σ A+B < 5 (R) = A B
Relational Algebra on Bags Why Bags? 13 14 A bag (or multiset ) is like a set, but an element may appear more than once. Example: {1,2,1,3} is a bag. Example: {1,2,3} is also a bag that happens to be a
More informationPlan of the lecture. G53RDB: Theory of Relational Databases Lecture 2. More operations: renaming. Previous lecture. Renaming.
Plan of the lecture G53RDB: Theory of Relational Lecture 2 Natasha Alechina chool of Computer cience & IT nza@cs.nott.ac.uk Renaming Joins Definability of intersection Division ome properties of relational
More informationInformation Systems for Engineers. Exercise 5. ETH Zurich, Fall Semester Hand-out Due
Information Systems for Engineers Exercise 5 ETH Zurich, Fall Semester 2017 Hand-out 27.10.2017 Due 03.11.2017 Reading material: Chapter 2.4 in [1]. Lecture slides 4. 1. Given the two tables below, write
More informationRelational-Database Design
C H A P T E R 7 Relational-Database Design Exercises 7.2 Answer: A decomposition {R 1, R 2 } is a lossless-join decomposition if R 1 R 2 R 1 or R 1 R 2 R 2. Let R 1 =(A, B, C), R 2 =(A, D, E), and R 1
More informationCS 347 Parallel and Distributed Data Processing
CS 347 Parallel and Distributed Data Processing Spring 2016 Notes 3: Query Processing Query Processing Decomposition Localization Optimization CS 347 Notes 3 2 Decomposition Same as in centralized system
More informationRelational Database: Identities of Relational Algebra; Example of Query Optimization
Relational Database: Identities of Relational Algebra; Example of Query Optimization Greg Plaxton Theory in Programming Practice, Fall 2005 Department of Computer Science University of Texas at Austin
More informationCS 347 Distributed Databases and Transaction Processing Notes03: Query Processing
CS 347 Distributed Databases and Transaction Processing Notes03: Query Processing Hector Garcia-Molina Zoltan Gyongyi CS 347 Notes 03 1 Query Processing! Decomposition! Localization! Optimization CS 347
More informationQuery Processing. 3 steps: Parsing & Translation Optimization Evaluation
rela%onal algebra Query Processing 3 steps: Parsing & Translation Optimization Evaluation 30 Simple set of algebraic operations on relations Journey of a query SQL select from where Rela%onal algebra π
More informationComputational Logic. Relational Query Languages with Negation. Free University of Bozen-Bolzano, Werner Nutt
Computational Logic Free University of Bozen-Bolzano, 2010 Werner Nutt (Slides adapted from Thomas Eiter and Leonid Libkin) Computational Logic 1 Queries with All Who are the directors whose movies are
More informationINTRODUCTION TO RELATIONAL DATABASE SYSTEMS
INTRODUCTION TO RELATIONAL DATABASE SYSTEMS DATENBANKSYSTEME 1 (INF 3131) Torsten Grust Universität Tübingen Winter 2017/18 1 THE RELATIONAL ALGEBRA The Relational Algebra (RA) is a query language for
More informationcse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska
cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska LECTURE 1 Course Web Page www3.cs.stonybrook.edu/ cse303 The webpage contains: lectures notes slides; very detailed solutions to
More informationDiscrete Mathematics & Mathematical Reasoning Course Overview
Discrete Mathematics & Mathematical Reasoning Course Overview Colin Stirling Informatics Colin Stirling (Informatics) Discrete Mathematics Today 1 / 19 Teaching staff Lecturers: Colin Stirling, first half
More informationRelational Algebra & Calculus
Relational Algebra & Calculus Yanlei Diao UMass Amherst Slides Courtesy of R. Ramakrishnan and J. Gehrke 1 Outline v Conceptual Design: ER model v Logical Design: ER to relational model v Querying and
More informationProving simple set properties...
Proving simple set properties... Part 1: Some examples of proofs over sets Fall 2013 Proving simple set properties... Fall 2013 1 / 17 Introduction Overview: Learning outcomes In this session we will...
More informationCS2742 midterm test 2 study sheet. Boolean circuits: Predicate logic:
x NOT ~x x y AND x /\ y x y OR x \/ y Figure 1: Types of gates in a digital circuit. CS2742 midterm test 2 study sheet Boolean circuits: Boolean circuits is a generalization of Boolean formulas in which
More informationCS54100: Database Systems
CS54100: Database Systems Relational Algebra 3 February 2012 Prof. Walid Aref Core Relational Algebra A small set of operators that allow us to manipulate relations in limited but useful ways. The operators
More informationDatabase Applications (15-415)
Database Applications (15-415) Relational Calculus Lecture 5, January 27, 2014 Mohammad Hammoud Today Last Session: Relational Algebra Today s Session: Relational algebra The division operator and summary
More informationEquivalence and Implication
Equivalence and Alice E. Fischer CSCI 1166 Discrete Mathematics for Computing February 7 8, 2018 Alice E. Fischer Laws of Logic... 1/33 1 Logical Equivalence Contradictions and Tautologies 2 3 4 Necessary
More informationCorrelated subqueries. Query Optimization. Magic decorrelation. COUNT bug. Magic example (slide 2) Magic example (slide 1)
Correlated subqueries Query Optimization CPS Advanced Database Systems SELECT CID FROM Course Executing correlated subquery is expensive The subquery is evaluated once for every CPS course Decorrelate!
More informationReview CHAPTER. 2.1 Definitions in Chapter Sample Exam Questions. 2.1 Set; Element; Member; Universal Set Partition. 2.
CHAPTER 2 Review 2.1 Definitions in Chapter 2 2.1 Set; Element; Member; Universal Set 2.2 Subset 2.3 Proper Subset 2.4 The Empty Set, 2.5 Set Equality 2.6 Cardinality; Infinite Set 2.7 Complement 2.8 Intersection
More informationDatabases Lecture 8. Timothy G. Griffin. Computer Laboratory University of Cambridge, UK. Databases, Lent 2009
Databases Lecture 8 Timothy G. Griffin Computer Laboratory University of Cambridge, UK Databases, Lent 2009 T. Griffin (cl.cam.ac.uk) Databases Lecture 8 DB 2009 1 / 15 Lecture 08: Multivalued Dependencies
More informationSets. Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry. Fall 2007
Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Fall 2007 1 / 42 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 2.1, 2.2 of Rosen Introduction I Introduction
More informationRelational Algebra as non-distributive Lattice
Relational Algebra as non-distributive Lattice VADIM TROPASHKO Vadim.Tropashko@orcl.com We reduce the set of classic relational algebra operators to two binary operations: natural join and generalized
More informationCOMP9414: Artificial Intelligence Propositional Logic: Automated Reasoning
COMP9414, Monday 26 March, 2012 Propositional Logic 2 COMP9414: Artificial Intelligence Propositional Logic: Automated Reasoning Overview Proof systems (including soundness and completeness) Normal Forms
More informationRelational Algebra Part 1. Definitions.
.. Cal Poly pring 2016 CPE/CC 365 Introduction to Database ystems Alexander Dekhtyar Eriq Augustine.. elational Algebra Notation, T,,... relations. t, t 1, t 2,... tuples of relations. t (n tuple with
More informationSets. Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry. Spring 2006
Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Spring 2006 1 / 1 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 1.6 1.7 of Rosen Introduction I We ve already
More information4. Sets The language of sets. Describing a Set. c Oksana Shatalov, Fall Set-builder notation (a more precise way of describing a set)
c Oksana Shatalov, Fall 2018 1 4. Sets 4.1. The language of sets Set Terminology and Notation Set is a well-defined collection of objects. Elements are objects or members of the set. Describing a Set Roster
More informationLecture : Set Theory and Logic
Lecture : Dr. Department of Mathematics Lovely Professional University Punjab, India October 18, 2014 Outline Contrapositive and Converse 1 Contrapositive and Converse 2 3 4 5 Contrapositive and Converse
More informationLogic and Databases. Phokion G. Kolaitis. UC Santa Cruz & IBM Research Almaden. Lecture 4 Part 1
Logic and Databases Phokion G. Kolaitis UC Santa Cruz & IBM Research Almaden Lecture 4 Part 1 1 Thematic Roadmap Logic and Database Query Languages Relational Algebra and Relational Calculus Conjunctive
More informationPredicate Calculus lecture 1
Predicate Calculus lecture 1 Section 1.3 Limitation of Propositional Logic Consider the following reasoning All cats have tails Gouchi is a cat Therefore, Gouchi has tail. MSU/CSE 260 Fall 2009 1 MSU/CSE
More informationSet Theory. CSE 215, Foundations of Computer Science Stony Brook University
Set Theory CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 Set theory Abstract set theory is one of the foundations of mathematical thought Most mathematical
More informationAdvanced Engineering Mathematics Prof. Pratima Panigrahi Department of Mathematics Indian Institute of Technology, Kharagpur
Advanced Engineering Mathematics Prof. Pratima Panigrahi Department of Mathematics Indian Institute of Technology, Kharagpur Lecture No. # 02 Vector Spaces, Subspaces, linearly Dependent/Independent of
More informationIntroduction to Quantum Logic. Chris Heunen
Introduction to Quantum Logic Chris Heunen 1 / 28 Overview Boolean algebra Superposition Quantum logic Entanglement Quantum computation 2 / 28 Boolean algebra 3 / 28 Boolean algebra A Boolean algebra is
More informationCPSC 121: Models of Computation
CPSC 121: Models of Computation Unit 6 Rewriting Predicate Logic Statements Based on slides by Patrice Belleville and Steve Wolfman Coming Up Pre-class quiz #7 is due Wednesday October 25th at 9:00 pm.
More informationExercises 1 - Solutions
Exercises 1 - Solutions SAV 2013 1 PL validity For each of the following propositional logic formulae determine whether it is valid or not. If it is valid prove it, otherwise give a counterexample. Note
More informationWith Question/Answer Animations. Chapter 2
With Question/Answer Animations Chapter 2 Chapter Summary Sets The Language of Sets Set Operations Set Identities Functions Types of Functions Operations on Functions Sequences and Summations Types of
More informationSets. Alice E. Fischer. CSCI 1166 Discrete Mathematics for Computing Spring, Outline Sets An Algebra on Sets Summary
An Algebra on Alice E. Fischer CSCI 1166 Discrete Mathematics for Computing Spring, 2018 Alice E. Fischer... 1/37 An Algebra on 1 Definitions and Notation Venn Diagrams 2 An Algebra on 3 Alice E. Fischer...
More information1. SET 10/9/2013. Discrete Mathematics Fajrian Nur Adnan, M.CS
1. SET 10/9/2013 Discrete Mathematics Fajrian Nur Adnan, M.CS 1 Discrete Mathematics 1. Set and Logic 2. Relation 3. Function 4. Induction 5. Boolean Algebra and Number Theory MID 6. Graf dan Tree/Pohon
More informationDatabase Applications (15-415)
Database Applications (15-415) Relational Calculus Lecture 6, January 26, 2016 Mohammad Hammoud Today Last Session: Relational Algebra Today s Session: Relational calculus Relational tuple calculus Announcements:
More informationAlso, course highlights the importance of discrete structures towards simulation of a problem in computer science and engineering.
The objectives of Discrete Mathematical Structures are: To introduce a number of Discrete Mathematical Structures (DMS) found to be serving as tools even today in the development of theoretical computer
More informationOverview key concepts and terms (based on the textbook Chang 2006 and the practical manual)
Introduction Geo-information Science (GRS-10306) Overview key concepts and terms (based on the textbook 2006 and the practical manual) Introduction Chapter 1 Geographic information system (GIS) Geographically
More informationRelational Database Design
Relational Database Design Jan Chomicki University at Buffalo Jan Chomicki () Relational database design 1 / 16 Outline 1 Functional dependencies 2 Normal forms 3 Multivalued dependencies Jan Chomicki
More informationA set is an unordered collection of objects.
Section 2.1 Sets A set is an unordered collection of objects. the students in this class the chairs in this room The objects in a set are called the elements, or members of the set. A set is said to contain
More informationCOSC 430 Advanced Database Topics. Lecture 2: Relational Theory Haibo Zhang Computer Science, University of Otago
COSC 430 Advanced Database Topics Lecture 2: Relational Theory Haibo Zhang Computer Science, University of Otago Learning objectives and references You should be able to: define the elements of the relational
More informationCombinational Logic Design Principles
Combinational Logic Design Principles Switching algebra Doru Todinca Department of Computers Politehnica University of Timisoara Outline Introduction Switching algebra Axioms of switching algebra Theorems
More information7 RC Simulates RA. Lemma: For every RA expression E(A 1... A k ) there exists a DRC formula F with F V (F ) = {A 1,..., A k } and
7 RC Simulates RA. We now show that DRC (and hence TRC) is at least as expressive as RA. That is, given an RA expression E that mentions at most C, there is an equivalent DRC expression E that mentions
More informationIntroduction. Foundations of Computing Science. Pallab Dasgupta Professor, Dept. of Computer Sc & Engg INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR
1 Introduction Foundations of Computing Science Pallab Dasgupta Professor, Dept. of Computer Sc & Engg 2 Comments on Alan Turing s Paper "On Computable Numbers, with an Application to the Entscheidungs
More information5 Systems of Equations
Systems of Equations Concepts: Solutions to Systems of Equations-Graphically and Algebraically Solving Systems - Substitution Method Solving Systems - Elimination Method Using -Dimensional Graphs to Approximate
More informationCS589 Principles of DB Systems Fall 2008 Lecture 4e: Logic (Model-theoretic view of a DB) Lois Delcambre
CS589 Principles of DB Systems Fall 2008 Lecture 4e: Logic (Model-theoretic view of a DB) Lois Delcambre lmd@cs.pdx.edu 503 725-2405 Goals for today Review propositional logic (including truth assignment)
More informationSchedule. Today: Jan. 17 (TH) Jan. 24 (TH) Jan. 29 (T) Jan. 22 (T) Read Sections Assignment 2 due. Read Sections Assignment 3 due.
Schedule Today: Jan. 17 (TH) Relational Algebra. Read Chapter 5. Project Part 1 due. Jan. 22 (T) SQL Queries. Read Sections 6.1-6.2. Assignment 2 due. Jan. 24 (TH) Subqueries, Grouping and Aggregation.
More informationA Dichotomy. in in Probabilistic Databases. Joint work with Robert Fink. for Non-Repeating Queries with Negation Queries with Negation
Dichotomy for Non-Repeating Queries with Negation Queries with Negation in in Probabilistic Databases Robert Dan Olteanu Fink and Dan Olteanu Joint work with Robert Fink Uncertainty in Computation Simons
More information2. Introduction to commutative rings (continued)
2. Introduction to commutative rings (continued) 2.1. New examples of commutative rings. Recall that in the first lecture we defined the notions of commutative rings and field and gave some examples of
More informationExpiration of Historical Databases
Expiration of Historical Databases (Extended Abstract) David Toman Department of Computer Science, University of Waterloo Waterloo, Ontario, Canada N2L 3G1 E-mail: david@uwaterloo.ca Abstract We present
More informationJUST THE MATHS UNIT NUMBER 1.3. ALGEBRA 3 (Indices and radicals (or surds)) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1 ALGEBRA (Indices and radicals (or surds)) by AJHobson 11 Indices 12 Radicals (or Surds) 1 Exercises 14 Answers to exercises UNIT 1 - ALGEBRA - INDICES AND RADICALS (or Surds)
More informationDiscrete Mathematics. (c) Marcin Sydow. Sets. Set operations. Sets. Set identities Number sets. Pair. Power Set. Venn diagrams
Contents : basic definitions and notation A set is an unordered collection of its elements (or members). The set is fully specified by its elements. Usually capital letters are used to name sets and lowercase
More informationYou are here! Query Processor. Recovery. Discussed here: DBMS. Task 3 is often called algebraic (or re-write) query optimization, while
Module 10: Query Optimization Module Outline 10.1 Outline of Query Optimization 10.2 Motivating Example 10.3 Equivalences in the relational algebra 10.4 Heuristic optimization 10.5 Explosion of search
More informationFinite Automata and Formal Languages TMV026/DIT321 LP4 2012
Finite Automata and Formal Languages TMV26/DIT32 LP4 22 Lecture 7 Ana Bove March 27th 22 Overview of today s lecture: Regular Expressions From FA to RE Regular Expressions Regular expressions (RE) are
More information3. Abstract Boolean Algebras
3. ABSTRACT BOOLEAN ALGEBRAS 123 3. Abstract Boolean Algebras 3.1. Abstract Boolean Algebra. Definition 3.1.1. An abstract Boolean algebra is defined as a set B containing two distinct elements 0 and 1,
More informationDatabase Design and Implementation
Database Design and Implementation CS 645 Data provenance Provenance provenance, n. The fact of coming from some particular source or quarter; origin, derivation [Oxford English Dictionary] Data provenance
More informationSet Theory Basics of Set Theory. 6.2 Properties of Sets and Element Argument. 6.3 Algebraic Proofs 6.4 Boolean Algebras.
9/6/17 Mustafa Jarrar: Lecture Notes in Discrete Mathematics. Birzeit University Palestine 2015 Set Theory 6.1. Basics of Set Theory 6.2 Properties of Sets and Element Argument 6.3 Algebraic Proofs 6.4
More informationProvenance Semirings. Todd Green Grigoris Karvounarakis Val Tannen. presented by Clemens Ley
Provenance Semirings Todd Green Grigoris Karvounarakis Val Tannen presented by Clemens Ley place of origin Provenance Semirings Todd Green Grigoris Karvounarakis Val Tannen presented by Clemens Ley place
More informationMovies Title Director Actor
Movies Title Director Actor The Trouble with Harry Hitchcock Gwenn The Trouble with Harry Hitchcock Forsythe The Trouble with Harry Hitchcock MacLaine The Trouble with Harry Hitchcock Hitchcock Cries and
More informationPlan of the lecture. G53RDB: Theory of Relational Databases Lecture 10. Logical consequence (implication) Implication problem for fds
Plan of the lecture G53RDB: Theory of Relational Databases Lecture 10 Natasha Alechina School of Computer Science & IT nza@cs.nott.ac.uk Logical implication for functional dependencies Armstrong closure.
More information1 Predicates and Quantifiers
1 Predicates and Quantifiers We have seen how to represent properties of objects. For example, B(x) may represent that x is a student at Bryn Mawr College. Here B stands for is a student at Bryn Mawr College
More informationSchema Refinement: Other Dependencies and Higher Normal Forms
Schema Refinement: Other Dependencies and Higher Normal Forms Spring 2018 School of Computer Science University of Waterloo Databases CS348 (University of Waterloo) Higher Normal Forms 1 / 14 Outline 1
More informationCHAPTER 1. Preliminaries. 1 Set Theory
CHAPTER 1 Preliminaries 1 et Theory We assume that the reader is familiar with basic set theory. In this paragraph, we want to recall the relevant definitions and fix the notation. Our approach to set
More information1 Alphabets and Languages
1 Alphabets and Languages Look at handout 1 (inference rules for sets) and use the rules on some examples like {a} {{a}} {a} {a, b}, {a} {{a}}, {a} {{a}}, {a} {a, b}, a {{a}}, a {a, b}, a {{a}}, a {a,
More informationSets McGraw-Hill Education
Sets A set is an 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 elements. The notation a A denotes that a is an element
More informationMAT 243 Test 1 SOLUTIONS, FORM A
t MAT 243 Test 1 SOLUTIONS, FORM A 1. [10 points] Rewrite the statement below in positive form (i.e., so that all negation symbols immediately precede a predicate). ( x IR)( y IR)((T (x, y) Q(x, y)) R(x,
More informationEECS-3421a: Test #2 Electrical Engineering & Computer Science York University
18 November 2015 EECS-3421a: Test #2 1 of 16 EECS-3421a: Test #2 Electrical Engineering & Computer Science York University Family Name: Given Name: Student#: CSE Account: Instructor: Parke Godfrey Exam
More informationLattice Theory Lecture 5. Completions
Lattice Theory Lecture 5 Completions John Harding New Mexico State University www.math.nmsu.edu/ JohnHarding.html jharding@nmsu.edu Toulouse, July 2017 Completions Definition A completion of a poset P
More informationEECS 1028 M: Discrete Mathematics for Engineers
EECS 1028 M: Discrete Mathematics for Engineers Suprakash Datta Office: LAS 3043 Course page: http://www.eecs.yorku.ca/course/1028 Also on Moodle S. Datta (York Univ.) EECS 1028 W 18 1 / 24 Course policies
More informationCSE 20 DISCRETE MATH. Winter
CSE 20 DISCRETE MATH Winter 2017 http://cseweb.ucsd.edu/classes/wi17/cse20-ab/ Today's learning goals Evaluate which proof technique(s) is appropriate for a given proposition Direct proof Proofs by contraposition
More informationMATHEMATICS Lecture. 4 Chapter.8 TECHNIQUES OF INTEGRATION By Dr. Mohammed Ramidh
MATHEMATICS Lecture. 4 Chapter.8 TECHNIQUES OF INTEGRATION By TECHNIQUES OF INTEGRATION OVERVIEW The Fundamental Theorem connects antiderivatives and the definite integral. Evaluating the indefinite integral,
More informationProofs. Chapter 2 P P Q Q
Chapter Proofs In this chapter we develop three methods for proving a statement. To start let s suppose the statement is of the form P Q or if P, then Q. Direct: This method typically starts with P. Then,
More informationLogic, Sets, and Proofs
Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Operators. A logical statement is a mathematical statement that can be assigned a value either true or false.
More informationCAIM: Cerca i Anàlisi d Informació Massiva
1 / 21 CAIM: Cerca i Anàlisi d Informació Massiva FIB, Grau en Enginyeria Informàtica Slides by Marta Arias, José Balcázar, Ricard Gavaldá Department of Computer Science, UPC Fall 2016 http://www.cs.upc.edu/~caim
More informationCMPS Advanced Database Systems. Dr. Chengwei Lei CEECS California State University, Bakersfield
CMPS 4420 Advanced Database Systems Dr. Chengwei Lei CEECS California State University, Bakersfield CHAPTER 15 Relational Database Design Algorithms and Further Dependencies Slide 15-2 Chapter Outline
More informationMathematical Preliminaries. Sipser pages 1-28
Mathematical Preliminaries Sipser pages 1-28 Mathematical Preliminaries This course is about the fundamental capabilities and limitations of computers. It has 3 parts 1. Automata Models of computation
More informationExercise Sheet 1: Relational Algebra David Carral, Markus Krötzsch Database Theory, 17 April, Summer Term 2018
Exercise Sheet 1: Relational Algebra David Carral, Markus Krötzsch Database Theory, 17 April, Summer Term 2018 Exercise 1.1. Consider a cinema database with tables of the following form (adapted from a
More informationTutorial on Axiomatic Set Theory. Javier R. Movellan
Tutorial on Axiomatic Set Theory Javier R. Movellan Intuitively we think of sets as collections of elements. The crucial part of this intuitive concept is that we are willing to treat sets as entities
More informationRelations on Hypergraphs
Relations on Hypergraphs John Stell School of Computing, University of Leeds RAMiCS 13 Cambridge, 17th September 2012 Relations on a Set Boolean algebra Converse R = R Complement R = R Composition & residuation
More informationOn Generalization of Fuzzy Connectives
n Generalization of Fuzzy Connectives Imre J. Rudas udapest Tech Doberdó út 6, H-1034 udapest, Hungary rudas@bmf.hu bstract: In real applications of fuzzy logic the properties of aggregation operators
More informationmat.haus project Zurab Janelidze s Lectures on UNIVERSE OF SETS last updated 18 May 2017 Stellenbosch University 2017
Zurab Janelidze s Lectures on UNIVERSE OF SETS last updated 18 May 2017 Stellenbosch University 2017 Contents 1. Axiom-free universe of sets 1 2. Equality of sets and the empty set 2 3. Comprehension and
More informationSelected solutions to Discrete Mathematics and Functional Programming
Selected solutions to Discrete Mathematics and Functional Programming Thomas VanDrunen August 13, 2013 This document is to provide a resource for students studying Discrete Mathematics and Functional Programming
More informationTheory of Computation
Theory of Computation Prof. Michael Mascagni Florida State University Department of Computer Science 1 / 33 This course aims to cover... the development of computability theory using an extremely simple
More informationChapter 2 Sets, Relations and Functions
Chapter 2 Sets, Relations and Functions Key Topics Sets Set Operations Russell s Paradox Relations Composition of Relations Reflexive, Symmetric and Transitive Relations Functions Partial and Total Functions
More informationCSE 544 Principles of Database Management Systems
CSE 544 Principles of Database Management Systems Lecture 3 Schema Normalization CSE 544 - Winter 2018 1 Announcements Project groups due on Friday First review due on Tuesday (makeup lecture) Run git
More informationFoundations of Mathematics Worksheet 2
Foundations of Mathematics Worksheet 2 L. Pedro Poitevin June 24, 2007 1. What are the atomic truth assignments on {a 1,..., a n } that satisfy: (a) The proposition p = ((a 1 a 2 ) (a 2 a 3 ) (a n 1 a
More informationLecture Notes on Metric Spaces
Lecture Notes on Metric Spaces Math 117: Summer 2007 John Douglas Moore Our goal of these notes is to explain a few facts regarding metric spaces not included in the first few chapters of the text [1],
More informationDatabase Design and Normalization
Database Design and Normalization Chapter 11 (Week 12) EE562 Slides and Modified Slides from Database Management Systems, R. Ramakrishnan 1 1NF FIRST S# Status City P# Qty S1 20 London P1 300 S1 20 London
More informationSection 2.2 Set Operations. Propositional calculus and set theory are both instances of an algebraic system called a. Boolean Algebra.
Section 2.2 Set Operations Propositional calculus and set theory are both instances of an algebraic system called a Boolean Algebra. The operators in set theory are defined in terms of the corresponding
More informationCompiling Techniques
Lecture 11: Introduction to 13 November 2015 Table of contents 1 Introduction Overview The Backend The Big Picture 2 Code Shape Overview Introduction Overview The Backend The Big Picture Source code FrontEnd
More informationSets. Introduction to Set Theory ( 2.1) Basic notations for sets. Basic properties of sets CMSC 302. Vojislav Kecman
Introduction to Set Theory ( 2.1) VCU, Department of Computer Science CMSC 302 Sets Vojislav Kecman A set is a new type of structure, representing an unordered collection (group, plurality) of zero or
More informationSection 4.4 Functions. CS 130 Discrete Structures
Section 4.4 Functions CS 130 Discrete Structures Function Definitions Let S and T be sets. A function f from S to T, f: S T, is a subset of S x T where each member of S appears exactly once as the first
More information1. Propositional Calculus
1. Propositional Calculus Some notes for Math 601, Fall 2010 based on Elliott Mendelson, Introduction to Mathematical Logic, Fifth edition, 2010, Chapman & Hall. 2. Syntax ( grammar ). 1.1, p. 1. Given:
More informationRelational Algebra 2. Week 5
Relational Algebra 2 Week 5 Relational Algebra (So far) Basic operations: Selection ( σ ) Selects a subset of rows from relation. Projection ( π ) Deletes unwanted columns from relation. Cross-product
More information