Relational Algebra
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2.1 the Algebra of Sets
Chapter 2 I Abstract Algebra 83 part of abstract algebra, sets are fundamental to all areas of mathematics and we need to establish a precise language for sets. We also explore operations on sets and relations between sets, developing an “algebra of sets” that strongly resembles aspects of the algebra of sentential logic. In addition, as we discussed in chapter 1, a fundamental goal in mathematics is crafting articulate, thorough, convincing, and insightful arguments for the truth of mathematical statements. We continue the development of theorem-proving and proof-writing skills in the context of basic set theory. After exploring the algebra of sets, we study two number systems denoted Zn and U(n) that are closely related to the integers. Our approach is based on a widely used strategy of mathematicians: we work with specific examples and look for general patterns. This study leads to the definition of modified addition and multiplication operations on certain finite subsets of the integers. We isolate key axioms, or properties, that are satisfied by these and many other number systems and then examine number systems that share the “group” properties of the integers. Finally, we consider an application of this mathematics to check digit schemes, which have become increasingly important for the success of business and telecommunications in our technologically based society. Through the study of these topics, we engage in a thorough introduction to abstract algebra from the perspective of the mathematician— working with specific examples to identify key abstract properties common to diverse and interesting mathematical systems. 2.1 The Algebra of Sets Intuitively, a set is a “collection” of objects known as “elements.” But in the early 1900’s, a radical transformation occurred in mathematicians’ understanding of sets when the British philosopher Bertrand Russell identified a fundamental paradox inherent in this intuitive notion of a set (this paradox is discussed in exercises 66–70 at the end of this section). -
The Five Fundamental Operations of Mathematics: Addition, Subtraction
The five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms Kenneth A. Ribet UC Berkeley Trinity University March 31, 2008 Kenneth A. Ribet Five fundamental operations This talk is about counting, and it’s about solving equations. Counting is a very familiar activity in mathematics. Many universities teach sophomore-level courses on discrete mathematics that turn out to be mostly about counting. For example, we ask our students to find the number of different ways of constituting a bag of a dozen lollipops if there are 5 different flavors. (The answer is 1820, I think.) Kenneth A. Ribet Five fundamental operations Solving equations is even more of a flagship activity for mathematicians. At a mathematics conference at Sundance, Robert Redford told a group of my colleagues “I hope you solve all your equations”! The kind of equations that I like to solve are Diophantine equations. Diophantus of Alexandria (third century AD) was Robert Redford’s kind of mathematician. This “father of algebra” focused on the solution to algebraic equations, especially in contexts where the solutions are constrained to be whole numbers or fractions. Kenneth A. Ribet Five fundamental operations Here’s a typical example. Consider the equation y 2 = x3 + 1. In an algebra or high school class, we might graph this equation in the plane; there’s little challenge. But what if we ask for solutions in integers (i.e., whole numbers)? It is relatively easy to discover the solutions (0; ±1), (−1; 0) and (2; ±3), and Diophantus might have asked if there are any more. -
Database Concepts 7Th Edition
Database Concepts 7th Edition David M. Kroenke • David J. Auer Online Appendix E SQL Views, SQL/PSM and Importing Data Database Concepts SQL Views, SQL/PSM and Importing Data Appendix E All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. Appendix E — 10 9 8 7 6 5 4 3 2 1 E-2 Database Concepts SQL Views, SQL/PSM and Importing Data Appendix E Appendix Objectives • To understand the reasons for using SQL views • To use SQL statements to create and query SQL views • To understand SQL/Persistent Stored Modules (SQL/PSM) • To create and use SQL user-defined functions • To import Microsoft Excel worksheet data into a database What is the Purpose of this Appendix? In Chapter 3, we discussed SQL in depth. We discussed two basic categories of SQL statements: data definition language (DDL) statements, which are used for creating tables, relationships, and other structures, and data manipulation language (DML) statements, which are used for querying and modifying data. In this appendix, which should be studied immediately after Chapter 3, we: • Describe and illustrate SQL views, which extend the DML capabilities of SQL. • Describe and illustrate SQL Persistent Stored Modules (SQL/PSM), and create user-defined functions. • Describe and use DBMS data import techniques to import Microsoft Excel worksheet data into a database. E-3 Database Concepts SQL Views, SQL/PSM and Importing Data Appendix E Creating SQL Views An SQL view is a virtual table that is constructed from other tables or views. -
A Quick Algebra Review
A Quick Algebra Review 1. Simplifying Expressions 2. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Exponents 9. Quadratics 10. Rationals 11. Radicals Simplifying Expressions An expression is a mathematical “phrase.” Expressions contain numbers and variables, but not an equal sign. An equation has an “equal” sign. For example: Expression: Equation: 5 + 3 5 + 3 = 8 x + 3 x + 3 = 8 (x + 4)(x – 2) (x + 4)(x – 2) = 10 x² + 5x + 6 x² + 5x + 6 = 0 x – 8 x – 8 > 3 When we simplify an expression, we work until there are as few terms as possible. This process makes the expression easier to use, (that’s why it’s called “simplify”). The first thing we want to do when simplifying an expression is to combine like terms. For example: There are many terms to look at! Let’s start with x². There Simplify: are no other terms with x² in them, so we move on. 10x x² + 10x – 6 – 5x + 4 and 5x are like terms, so we add their coefficients = x² + 5x – 6 + 4 together. 10 + (-5) = 5, so we write 5x. -6 and 4 are also = x² + 5x – 2 like terms, so we can combine them to get -2. Isn’t the simplified expression much nicer? Now you try: x² + 5x + 3x² + x³ - 5 + 3 [You should get x³ + 4x² + 5x – 2] Order of Operations PEMDAS – Please Excuse My Dear Aunt Sally, remember that from Algebra class? It tells the order in which we can complete operations when solving an equation. -
7.2 Binary Operators Closure
last edited April 19, 2016 7.2 Binary Operators A precise discussion of symmetry benefits from the development of what math- ematicians call a group, which is a special kind of set we have not yet explicitly considered. However, before we define a group and explore its properties, we reconsider several familiar sets and some of their most basic features. Over the last several sections, we have considered many di↵erent kinds of sets. We have considered sets of integers (natural numbers, even numbers, odd numbers), sets of rational numbers, sets of vertices, edges, colors, polyhedra and many others. In many of these examples – though certainly not in all of them – we are familiar with rules that tell us how to combine two elements to form another element. For example, if we are dealing with the natural numbers, we might considered the rules of addition, or the rules of multiplication, both of which tell us how to take two elements of N and combine them to give us a (possibly distinct) third element. This motivates the following definition. Definition 26. Given a set S,abinary operator ? is a rule that takes two elements a, b S and manipulates them to give us a third, not necessarily distinct, element2 a?b. Although the term binary operator might be new to us, we are already familiar with many examples. As hinted to earlier, the rule for adding two numbers to give us a third number is a binary operator on the set of integers, or on the set of rational numbers, or on the set of real numbers. -
Rules for Matrix Operations
Math 2270 - Lecture 8: Rules for Matrix Operations Dylan Zwick Fall 2012 This lecture covers section 2.4 of the textbook. 1 Matrix Basix Most of this lecture is about formalizing rules and operations that we’ve already been using in the class up to this point. So, it should be mostly a review, but a necessary one. If any of this is new to you please make sure you understand it, as it is the foundation for everything else we’ll be doing in this course! A matrix is a rectangular array of numbers, and an “m by n” matrix, also written rn x n, has rn rows and n columns. We can add two matrices if they are the same shape and size. Addition is termwise. We can also mul tiply any matrix A by a constant c, and this multiplication just multiplies every entry of A by c. For example: /2 3\ /3 5\ /5 8 (34 )+( 10 Hf \i 2) \\2 3) \\3 5 /1 2\ /3 6 3 3 ‘ = 9 12 1 I 1 2 4) \6 12 1 Moving on. Matrix multiplication is more tricky than matrix addition, because it isn’t done termwise. In fact, if two matrices have the same size and shape, it’s not necessarily true that you can multiply them. In fact, it’s only true if that shape is square. In order to multiply two matrices A and B to get AB the number of columns of A must equal the number of rows of B. So, we could not, for example, multiply a 2 x 3 matrix by a 2 x 3 matrix. -
Relational Algebra
Relational Algebra Instructor: Shel Finkelstein Reference: A First Course in Database Systems, 3rd edition, Chapter 2.4 – 2.6, plus Query Execution Plans Important Notices • Midterm with Answers has been posted on Piazza. – Midterm will be/was reviewed briefly in class on Wednesday, Nov 8. – Grades were posted on Canvas on Monday, Nov 13. • Median was 83; no curve. – Exam will be returned in class on Nov 13 and Nov 15. • Please send email if you want “cheat sheet” back. • Lab3 assignment was posted on Sunday, Nov 5, and is due by Sunday, Nov 19, 11:59pm. – Lab3 has lots of parts (some hard), and is worth 13 points. – Please attend Labs to get help with Lab3. What is a Data Model? • A data model is a mathematical formalism that consists of three parts: 1. A notation for describing and representing data (structure of the data) 2. A set of operations for manipulating data. 3. A set of constraints on the data. • What is the associated query language for the relational data model? Two Query Languages • Codd proposed two different query languages for the relational data model. – Relational Algebra • Queries are expressed as a sequence of operations on relations. • Procedural language. – Relational Calculus • Queries are expressed as formulas of first-order logic. • Declarative language. • Codd’s Theorem: The Relational Algebra query language has the same expressive power as the Relational Calculus query language. Procedural vs. Declarative Languages • Procedural program – The program is specified as a sequence of operations to obtain the desired the outcome. I.e., how the outcome is to be obtained. -
Basic Concepts of Set Theory, Functions and Relations 1. Basic
Ling 310, adapted from UMass Ling 409, Partee lecture notes March 1, 2006 p. 1 Basic Concepts of Set Theory, Functions and Relations 1. Basic Concepts of Set Theory........................................................................................................................1 1.1. Sets and elements ...................................................................................................................................1 1.2. Specification of sets ...............................................................................................................................2 1.3. Identity and cardinality ..........................................................................................................................3 1.4. Subsets ...................................................................................................................................................4 1.5. Power sets .............................................................................................................................................4 1.6. Operations on sets: union, intersection...................................................................................................4 1.7 More operations on sets: difference, complement...................................................................................5 1.8. Set-theoretic equalities ...........................................................................................................................5 Chapter 2. Relations and Functions ..................................................................................................................6 -
Binary Operations
4-22-2007 Binary Operations Definition. A binary operation on a set X is a function f : X × X → X. In other words, a binary operation takes a pair of elements of X and produces an element of X. It’s customary to use infix notation for binary operations. Thus, rather than write f(a, b) for the binary operation acting on elements a, b ∈ X, you write afb. Since all those letters can get confusing, it’s also customary to use certain symbols — +, ·, ∗ — for binary operations. Thus, f(a, b) becomes (say) a + b or a · b or a ∗ b. Example. Addition is a binary operation on the set Z of integers: For every pair of integers m, n, there corresponds an integer m + n. Multiplication is also a binary operation on the set Z of integers: For every pair of integers m, n, there corresponds an integer m · n. However, division is not a binary operation on the set Z of integers. For example, if I take the pair 3 (3, 0), I can’t perform the operation . A binary operation on a set must be defined for all pairs of elements 0 from the set. Likewise, a ∗ b = (a random number bigger than a or b) does not define a binary operation on Z. In this case, I don’t have a function Z × Z → Z, since the output is ambiguously defined. (Is 3 ∗ 5 equal to 6? Or is it 117?) When a binary operation occurs in mathematics, it usually has properties that make it useful in con- structing abstract structures. -
How to Get Data from Oracle to Postgresql and Vice Versa Who We Are
How to get data from Oracle to PostgreSQL and vice versa Who we are The Company > Founded in 2010 > More than 70 specialists > Specialized in the Middleware Infrastructure > The invisible part of IT > Customers in Switzerland and all over Europe Our Offer > Consulting > Service Level Agreements (SLA) > Trainings > License Management How to get data from Oracle to PostgreSQL and vice versa 19.06.2020 Page 2 About me Daniel Westermann Principal Consultant Open Infrastructure Technology Leader +41 79 927 24 46 daniel.westermann[at]dbi-services.com @westermanndanie Daniel Westermann How to get data from Oracle to PostgreSQL and vice versa 19.06.2020 Page 3 How to get data from Oracle to PostgreSQL and vice versa Before we start We have a PostgreSQL user group in Switzerland! > https://www.swisspug.org Consider supporting us! How to get data from Oracle to PostgreSQL and vice versa 19.06.2020 Page 4 How to get data from Oracle to PostgreSQL and vice versa Before we start We have a PostgreSQL meetup group in Switzerland! > https://www.meetup.com/Switzerland-PostgreSQL-User-Group/ Consider joining us! How to get data from Oracle to PostgreSQL and vice versa 19.06.2020 Page 5 Agenda 1.Past, present and future 2.SQL/MED 3.Foreign data wrappers 4.Demo 5.Conclusion How to get data from Oracle to PostgreSQL and vice versa 19.06.2020 Page 6 Disclaimer This session is not about logical replication! If you are looking for this: > Data Replicator from DBPLUS > https://blog.dbi-services.com/real-time-replication-from-oracle-to-postgresql-using-data-replicator-from-dbplus/ -
Handling Missing Values in the SQL Procedure
Handling Missing Values in the SQL Procedure Danbo Yi, Abt Associates Inc., Cambridge, MA Lei Zhang, Domain Solutions Corp., Cambridge, MA non-missing numeric values. A missing ABSTRACT character value is expressed and treated as a string of blanks. Missing character values PROC SQL as a powerful database are always same no matter whether it is management tool provides many features expressed as one blank, or more than one available in the DATA steps and the blanks. Obviously, missing character values MEANS, TRANSPOSE, PRINT and SORT are not the smallest strings. procedures. If properly used, PROC SQL often results in concise solutions to data In SAS system, the way missing Date and manipulations and queries. PROC SQL DateTime values are expressed and treated follows most of the guidelines set by the is similar to missing numeric values. American National Standards Institute (ANSI) in its implementation of SQL. This paper will cover following topics. However, it is not fully compliant with the 1. Missing Values and Expression current ANSI Standard for SQL, especially · Logic Expression for the missing values. PROC SQL uses · Arithmetic Expression SAS System convention to express and · String Expression handle the missing values, which is 2. Missing Values and Predicates significantly different from many ANSI- · IS NULL /IS MISSING compatible SQL databases such as Oracle, · [NOT] LIKE Sybase. In this paper, we summarize the · ALL, ANY, and SOME ways the PROC SQL handles the missing · [NOT] EXISTS values in a variety of situations. Topics 3. Missing Values and JOINs include missing values in the logic, · Inner Join arithmetic and string expression, missing · Left/Right Join values in the SQL predicates such as LIKE, · Full Join ANY, ALL, JOINs, missing values in the 4. -
Firebird 3 Windowing Functions
Firebird 3 Windowing Functions Firebird 3 Windowing Functions Author: Philippe Makowski IBPhoenix Email: pmakowski@ibphoenix Licence: Public Documentation License Date: 2011-11-22 Philippe Makowski - IBPhoenix - 2011-11-22 Firebird 3 Windowing Functions What are Windowing Functions? • Similar to classical aggregates but does more! • Provides access to set of rows from the current row • Introduced SQL:2003 and more detail in SQL:2008 • Supported by PostgreSQL, Oracle, SQL Server, Sybase and DB2 • Used in OLAP mainly but also useful in OLTP • Analysis and reporting by rankings, cumulative aggregates Philippe Makowski - IBPhoenix - 2011-11-22 Firebird 3 Windowing Functions Windowed Table Functions • Windowed table function • operates on a window of a table • returns a value for every row in that window • the value is calculated by taking into consideration values from the set of rows in that window • 8 new windowed table functions • In addition, old aggregate functions can also be used as windowed table functions • Allows calculation of moving and cumulative aggregate values. Philippe Makowski - IBPhoenix - 2011-11-22 Firebird 3 Windowing Functions A Window • Represents set of rows that is used to compute additionnal attributes • Based on three main concepts • partition • specified by PARTITION BY clause in OVER() • Allows to subdivide the table, much like GROUP BY clause • Without a PARTITION BY clause, the whole table is in a single partition • order • defines an order with a partition • may contain multiple order items • Each item includes