A Theory of Particular Sets, As It Uses Open Formulas Rather Than Sequents
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1 Elementary Set Theory
1 Elementary Set Theory Notation: fg enclose a set. f1; 2; 3g = f3; 2; 2; 1; 3g because a set is not defined by order or multiplicity. f0; 2; 4;:::g = fxjx is an even natural numberg because two ways of writing a set are equivalent. ; is the empty set. x 2 A denotes x is an element of A. N = f0; 1; 2;:::g are the natural numbers. Z = f:::; −2; −1; 0; 1; 2;:::g are the integers. m Q = f n jm; n 2 Z and n 6= 0g are the rational numbers. R are the real numbers. Axiom 1.1. Axiom of Extensionality Let A; B be sets. If (8x)x 2 A iff x 2 B then A = B. Definition 1.1 (Subset). Let A; B be sets. Then A is a subset of B, written A ⊆ B iff (8x) if x 2 A then x 2 B. Theorem 1.1. If A ⊆ B and B ⊆ A then A = B. Proof. Let x be arbitrary. Because A ⊆ B if x 2 A then x 2 B Because B ⊆ A if x 2 B then x 2 A Hence, x 2 A iff x 2 B, thus A = B. Definition 1.2 (Union). Let A; B be sets. The Union A [ B of A and B is defined by x 2 A [ B if x 2 A or x 2 B. Theorem 1.2. A [ (B [ C) = (A [ B) [ C Proof. Let x be arbitrary. x 2 A [ (B [ C) iff x 2 A or x 2 B [ C iff x 2 A or (x 2 B or x 2 C) iff x 2 A or x 2 B or x 2 C iff (x 2 A or x 2 B) or x 2 C iff x 2 A [ B or x 2 C iff x 2 (A [ B) [ C Definition 1.3 (Intersection). -
Domain and Range of a Function
4.1 Domain and Range of a Function How can you fi nd the domain and range of STATES a function? STANDARDS MA.8.A.1.1 MA.8.A.1.5 1 ACTIVITY: The Domain and Range of a Function Work with a partner. The table shows the number of adult and child tickets sold for a school concert. Input Number of Adult Tickets, x 01234 Output Number of Child Tickets, y 86420 The variables x and y are related by the linear equation 4x + 2y = 16. a. Write the equation in function form by solving for y. b. The domain of a function is the set of all input values. Find the domain of the function. Domain = Why is x = 5 not in the domain of the function? 1 Why is x = — not in the domain of the function? 2 c. The range of a function is the set of all output values. Find the range of the function. Range = d. Functions can be described in many ways. ● by an equation ● by an input-output table y 9 ● in words 8 ● by a graph 7 6 ● as a set of ordered pairs 5 4 Use the graph to write the function 3 as a set of ordered pairs. 2 1 , , ( , ) ( , ) 0 09321 45 876 x ( , ) , ( , ) , ( , ) 148 Chapter 4 Functions 2 ACTIVITY: Finding Domains and Ranges Work with a partner. ● Copy and complete each input-output table. ● Find the domain and range of the function represented by the table. 1 a. y = −3x + 4 b. y = — x − 6 2 x −2 −10 1 2 x 01234 y y c. -
Module 1 Lecture Notes
Module 1 Lecture Notes Contents 1.1 Identifying Functions.............................1 1.2 Algebraically Determining the Domain of a Function..........4 1.3 Evaluating Functions.............................6 1.4 Function Operations..............................7 1.5 The Difference Quotient...........................9 1.6 Applications of Function Operations.................... 10 1.7 Determining the Domain and Range of a Function Graphically.... 12 1.8 Reading the Graph of a Function...................... 14 1.1 Identifying Functions In previous classes, you should have studied a variety basic functions. For example, 1 p f(x) = 3x − 5; g(x) = 2x2 − 1; h(x) = ; j(x) = 5x + 2 x − 5 We will begin this course by studying functions and their properties. As the course progresses, we will study inverse, composite, exponential, logarithmic, polynomial and rational functions. Math 111 Module 1 Lecture Notes Definition 1: A relation is a correspondence between two variables. A relation can be ex- pressed through a set of ordered pairs, a graph, a table, or an equation. A set containing ordered pairs (x; y) defines y as a function of x if and only if no two ordered pairs in the set have the same x-coordinate. In other words, every input maps to exactly one output. We write y = f(x) and say \y is a function of x." For the function defined by y = f(x), • x is the independent variable (also known as the input) • y is the dependent variable (also known as the output) • f is the function name Example 1: Determine whether or not each of the following represents a function. Table 1.1 Chicken Name Egg Color Emma Turquoise Hazel Light Brown George(ia) Chocolate Brown Isabella White Yvonne Light Brown (a) The set of ordered pairs of the form (chicken name, egg color) shown in Table 1.1. -
Formal Construction of a Set Theory in Coq
Saarland University Faculty of Natural Sciences and Technology I Department of Computer Science Masters Thesis Formal Construction of a Set Theory in Coq submitted by Jonas Kaiser on November 23, 2012 Supervisor Prof. Dr. Gert Smolka Advisor Dr. Chad E. Brown Reviewers Prof. Dr. Gert Smolka Dr. Chad E. Brown Eidesstattliche Erklarung¨ Ich erklare¨ hiermit an Eides Statt, dass ich die vorliegende Arbeit selbststandig¨ verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet habe. Statement in Lieu of an Oath I hereby confirm that I have written this thesis on my own and that I have not used any other media or materials than the ones referred to in this thesis. Einverstandniserkl¨ arung¨ Ich bin damit einverstanden, dass meine (bestandene) Arbeit in beiden Versionen in die Bibliothek der Informatik aufgenommen und damit vero¨ffentlicht wird. Declaration of Consent I agree to make both versions of my thesis (with a passing grade) accessible to the public by having them added to the library of the Computer Science Department. Saarbrucken,¨ (Datum/Date) (Unterschrift/Signature) iii Acknowledgements First of all I would like to express my sincerest gratitude towards my advisor, Chad Brown, who supported me throughout this work. His extensive knowledge and insights opened my eyes to the beauty of axiomatic set theory and foundational mathematics. We spent many hours discussing the minute details of the various constructions and he taught me the importance of mathematical rigour. Equally important was the support of my supervisor, Prof. Smolka, who first introduced me to the topic and was there whenever a question arose. -
The Church-Turing Thesis in Quantum and Classical Computing∗
The Church-Turing thesis in quantum and classical computing∗ Petrus Potgietery 31 October 2002 Abstract The Church-Turing thesis is examined in historical context and a survey is made of current claims to have surpassed its restrictions using quantum computing | thereby calling into question the basis of the accepted solutions to the Entscheidungsproblem and Hilbert's tenth problem. 1 Formal computability The need for a formal definition of algorithm or computation became clear to the scientific community of the early twentieth century due mainly to two problems of Hilbert: the Entscheidungsproblem|can an algorithm be found that, given a statement • in first-order logic (for example Peano arithmetic), determines whether it is true in all models of a theory; and Hilbert's tenth problem|does there exist an algorithm which, given a Diophan- • tine equation, determines whether is has any integer solutions? The Entscheidungsproblem, or decision problem, can be traced back to Leibniz and was successfully and independently solved in the mid-1930s by Alonzo Church [Church, 1936] and Alan Turing [Turing, 1935]. Church defined an algorithm to be identical to that of a function computable by his Lambda Calculus. Turing, in turn, first identified an algorithm to be identical with a recursive function and later with a function computed by what is now known as a Turing machine1. It was later shown that the class of the recursive functions, Turing machines, and the Lambda Calculus define the same class of functions. The remarkable equivalence of these three definitions of disparate origin quite strongly supported the idea of this as an adequate definition of computability and the Entscheidungsproblem is generally considered to be solved. -
Turing Machines
Syllabus R09 Regulation FORMAL LANGUAGES & AUTOMATA THEORY Jaya Krishna, M.Tech, Asst. Prof. UNIT-VII TURING MACHINES Introduction Until now we have been interested primarily in simple classes of languages and the ways that they can be used for relatively constrainted problems, such as analyzing protocols, searching text, or parsing programs. We begin with an informal argument, using an assumed knowledge of C programming, to show that there are specific problems we cannot solve using a computer. These problems are called “undecidable”. We then introduce a respected formalism for computers called the Turing machine. While a Turing machine looks nothing like a PC, and would be grossly inefficient should some startup company decide to manufacture and sell them, the Turing machine long has been recognized as an accurate model for what any physical computing device is capable of doing. TURING MACHINE: The purpose of the theory of undecidable problems is not only to establish the existence of such problems – an intellectually exciting idea in its own right – but to provide guidance to programmers about what they might or might not be able to accomplish through programming. The theory has great impact when we discuss the problems that are decidable, may require large amount of time to solve them. These problems are called “intractable problems” tend to present greater difficulty to the programmer and system designer than do the undecidable problems. We need tools that will allow us to prove everyday questions undecidable or intractable. As a result we need to rebuild our theory of undecidability, based not on programs in C or another language, but based on a very simple model of a computer called the Turing machine. -
The Development of Mathematical Logic from Russell to Tarski: 1900–1935
The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu Richard Zach Calixto Badesa The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu (University of California, Berkeley) Richard Zach (University of Calgary) Calixto Badesa (Universitat de Barcelona) Final Draft—May 2004 To appear in: Leila Haaparanta, ed., The Development of Modern Logic. New York and Oxford: Oxford University Press, 2004 Contents Contents i Introduction 1 1 Itinerary I: Metatheoretical Properties of Axiomatic Systems 3 1.1 Introduction . 3 1.2 Peano’s school on the logical structure of theories . 4 1.3 Hilbert on axiomatization . 8 1.4 Completeness and categoricity in the work of Veblen and Huntington . 10 1.5 Truth in a structure . 12 2 Itinerary II: Bertrand Russell’s Mathematical Logic 15 2.1 From the Paris congress to the Principles of Mathematics 1900–1903 . 15 2.2 Russell and Poincar´e on predicativity . 19 2.3 On Denoting . 21 2.4 Russell’s ramified type theory . 22 2.5 The logic of Principia ......................... 25 2.6 Further developments . 26 3 Itinerary III: Zermelo’s Axiomatization of Set Theory and Re- lated Foundational Issues 29 3.1 The debate on the axiom of choice . 29 3.2 Zermelo’s axiomatization of set theory . 32 3.3 The discussion on the notion of “definit” . 35 3.4 Metatheoretical studies of Zermelo’s axiomatization . 38 4 Itinerary IV: The Theory of Relatives and Lowenheim’s¨ Theorem 41 4.1 Theory of relatives and model theory . 41 4.2 The logic of relatives . -
Domain and Range of an Inverse Function
Lesson 28 Domain and Range of an Inverse Function As stated in the previous lesson, when changing from a function to its inverse the inputs and outputs of the original function are switched. This is because when we find an inverse function, we are taking some original function and solving for its input 푥; so what used to be the input becomes the output, and what used to be the output becomes the input. −11 푓(푥) = Given to the left are the steps 1 + 3푥 to find the inverse of the −ퟏퟏ original function 푓. These 풇 = steps illustrates the changing ퟏ + ퟑ풙 of the inputs and the outputs (ퟏ + ퟑ풙) ∙ 풇 = −ퟏퟏ when going from a function to its inverse. We start out 풇 + ퟑ풙풇 = −ퟏퟏ with 푓 (the output) isolated and 푥 (the input) as part of ퟑ풙풇 = −ퟏퟏ − 풇 the expression, and we end up with 푥 isolated and 푓 as −ퟏퟏ − 풇 the input of the expression. 풙 = ퟑ풇 Keep in mind that once 푥 is isolated, we have basically −11 − 푥 푓−1(푥) = found the inverse function. 3푥 Since the inputs and outputs of a function are switched when going from the original function to its inverse, this means that the domain of the original function 푓 is the range of its inverse function 푓−1. This also means that the range of the original function 푓 is the domain of its inverse function 푓−1. In this lesson we will review how to find an inverse function (as shown above), and we will also review how to find the domain of a function (which we covered in Lesson 18). -
First-Order Logic
Chapter 5 First-Order Logic 5.1 INTRODUCTION In propositional logic, it is not possible to express assertions about elements of a structure. The weak expressive power of propositional logic accounts for its relative mathematical simplicity, but it is a very severe limitation, and it is desirable to have more expressive logics. First-order logic is a considerably richer logic than propositional logic, but yet enjoys many nice mathemati- cal properties. In particular, there are finitary proof systems complete with respect to the semantics. In first-order logic, assertions about elements of structures can be ex- pressed. Technically, this is achieved by allowing the propositional symbols to have arguments ranging over elements of structures. For convenience, we also allow symbols denoting functions and constants. Our study of first-order logic will parallel the study of propositional logic conducted in Chapter 3. First, the syntax of first-order logic will be defined. The syntax is given by an inductive definition. Next, the semantics of first- order logic will be given. For this, it will be necessary to define the notion of a structure, which is essentially the concept of an algebra defined in Section 2.4, and the notion of satisfaction. Given a structure M and a formula A, for any assignment s of values in M to the variables (in A), we shall define the satisfaction relation |=, so that M |= A[s] 146 5.2 FIRST-ORDER LANGUAGES 147 expresses the fact that the assignment s satisfies the formula A in M. The satisfaction relation |= is defined recursively on the set of formulae. -
Functions and Relations
Functions and relations • Relations • Set rules & symbols • Sets of numbers • Sets & intervals • Functions • Relations • Function notation • Hybrid functions • Hyperbola • Truncus • Square root • Circle • Inverse functions VCE Maths Methods - Functions & Relations 2 Relations • A relation is a rule that links two sets of numbers: the domain & range. • The domain of a relation is the set of the !rst elements of the ordered pairs (x values). • The range of a relation is the set of the second elements of the ordered pairs (y values). • (The range is a subset of the co-domain of the function.) • Some relations exist for all possible values of x. • Other relations have an implied domain, as the function is only valid for certain values of x. VCE Maths Methods - Functions & Relations 3 Set rules & symbols A = {1, 3, 5, 7, 9} B = {2, 4, 6, 8, 10} C = {2, 3, 5, 7} D = {4,8} (odd numbers) (even numbers) (prime numbers) (multiples of 4) • ∈ : element of a set. 3 ! A • ∉ : not an element of a set. 6 ! A • ∩ : intersection of two sets (in both B and C) B!C = {2} ∪ • : union of two sets (elements in set B or C) B!C = {2,3,4,5,6,7,8,10} • ⊂ : subset of a set (all elements in D are part of set B) D ! B • \ : exclusion (elements in set B but not in set C) B \C = {2,6,10} • Ø : empty set A!B =" VCE Maths Methods - Functions & Relations 4 Sets of numbers • The domain & range of a function are each a subset of a particular larger set of numbers. -
Frege and the Logic of Sense and Reference
FREGE AND THE LOGIC OF SENSE AND REFERENCE Kevin C. Klement Routledge New York & London Published in 2002 by Routledge 29 West 35th Street New York, NY 10001 Published in Great Britain by Routledge 11 New Fetter Lane London EC4P 4EE Routledge is an imprint of the Taylor & Francis Group Printed in the United States of America on acid-free paper. Copyright © 2002 by Kevin C. Klement All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical or other means, now known or hereafter invented, including photocopying and recording, or in any infomration storage or retrieval system, without permission in writing from the publisher. 10 9 8 7 6 5 4 3 2 1 Library of Congress Cataloging-in-Publication Data Klement, Kevin C., 1974– Frege and the logic of sense and reference / by Kevin Klement. p. cm — (Studies in philosophy) Includes bibliographical references and index ISBN 0-415-93790-6 1. Frege, Gottlob, 1848–1925. 2. Sense (Philosophy) 3. Reference (Philosophy) I. Title II. Studies in philosophy (New York, N. Y.) B3245.F24 K54 2001 12'.68'092—dc21 2001048169 Contents Page Preface ix Abbreviations xiii 1. The Need for a Logical Calculus for the Theory of Sinn and Bedeutung 3 Introduction 3 Frege’s Project: Logicism and the Notion of Begriffsschrift 4 The Theory of Sinn and Bedeutung 8 The Limitations of the Begriffsschrift 14 Filling the Gap 21 2. The Logic of the Grundgesetze 25 Logical Language and the Content of Logic 25 Functionality and Predication 28 Quantifiers and Gothic Letters 32 Roman Letters: An Alternative Notation for Generality 38 Value-Ranges and Extensions of Concepts 42 The Syntactic Rules of the Begriffsschrift 44 The Axiomatization of Frege’s System 49 Responses to the Paradox 56 v vi Contents 3. -
A Functional Hitchhiker's Guide to Hereditarily Finite Sets, Ackermann
A Functional Hitchhiker’s Guide to Hereditarily Finite Sets, Ackermann Encodings and Pairing Functions – unpublished draft – Paul Tarau Department of Computer Science and Engineering University of North Texas [email protected] Abstract interest in various fields, from representing structured data The paper is organized as a self-contained literate Haskell in databases (Leontjev and Sazonov 2000) to reasoning with program that implements elements of an executable fi- sets and set constraints in a Logic Programming framework nite set theory with focus on combinatorial generation and (Dovier et al. 2000; Piazza and Policriti 2004; Dovier et al. arithmetic encodings. The code, tested under GHC 6.6.1, 2001). is available at http://logic.csci.unt.edu/tarau/ The Universe of Hereditarily Finite Sets is built from the research/2008/fSET.zip. empty set (or a set of Urelements) by successively applying We introduce ranking and unranking functions generaliz- powerset and set union operations. A surprising bijection, ing Ackermann’s encoding to the universe of Hereditarily Fi- discovered by Wilhelm Ackermann in 1937 (Ackermann nite Sets with Urelements. Then we build a lazy enumerator 1937; Abian and Lamacchia 1978; Kaye and Wong 2007) for Hereditarily Finite Sets with Urelements that matches the from Hereditarily Finite Sets to Natural Numbers, was the unranking function provided by the inverse of Ackermann’s original trigger for our work on building in a mathematically encoding and we describe functors between them resulting elegant programming language, a concise and executable in arithmetic encodings for powersets, hypergraphs, ordinals hereditarily finite set theory. The arbitrary size of the data and choice functions.