From Lagrange to Frege: Functions and Expressions Marco Panza
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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. -
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. -
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. -
Commencement1976.Pdf (4.717Mb)
1976 Digitized by the Internet Archive in 2012 with funding from LYRASIS Members and Sloan Foundation http://archive.org/details/commencement1976 ORDER OF PROCESSION MARSHALS STANLEY CORRSIN JOHN W. GRYDER MATTHEW A. CRENSON WILLIAM H. HUGGINS ELAINE C. DAVIS ROBERT A. LYSTAD HANS GOEDICKE EVANGELOS N. MOUDRIANAKIS ARCHIE GOLDEN EVERETT SCHILLER GERALD S. GOTTERER JOHN P. YOUNG THE GRADUATES MARSHALS ROBERT B. POND OREST RANUM THE DEANS MEMBERS OF THE SOCIETY OF SCHOLARS OFFICERS OF THE UNIVERSITY THE TRUSTEES * MARSHALS BROWN L. MURR FRANCIS E. ROURKE THE FACULTIES * CHIEF MARSHAL RICHARD A. MACKSEY THE CHAPLAINS THE RECIPIENT OF THE MILTON STOVER EISENHOWER MEDAL FOR DISTINGUISHED SERVICE THE PRESENTOR OF THE RECIPIENT OF THE MILTON STOVER EISENHOWER MEDAL FOR DISTINGUISHED SERVICE THE HONORARY DEGREE CANDIDATES THE PROVOST OF THE UNIVERSITY THE PRESIDENT EMERITUS OF THE UNIVERSITY THE CHAIRMAN OF THE BOARD OF TRUSTEES THE PRESIDENT OF THE UNIVERSITY ORDER OF EVENTS STEVEN MULLER President of the University, presiding * * * FANFARE PROCESSIONAL The audience is requested to stand as the Academic Procession moves into the area and to remain standing after the Invocation. " " Earle of Oxford's Marche William Byrd The Peabody Wind Ensemble Richard Higgins, Director * INVOCATION REV. CHESTER WICKWIRE Chaplain, The Johns Hopkins University THE NATIONAL ANTHEM GREETINGS ROBERT D. H. HARVEY Chairman of the Board of Trustees PRESENTATION OF THE RECIPIENT FOR THE MILTON STOVER EISENHOWER MEDAL FOR DISTINGUISHED SERVICE HELEN B. TAUSSIG PRESENTED BY RICHARD S. ROSS Vice President for the Health Divisions and Dean, School of Medicine * PRESENTATION OF NEW MEMBERS OF THE SOCIETY OF SCHOLARS LEROY E. -
A Theory of Particular Sets, As It Uses Open Formulas Rather Than Sequents
A Theory of Particular Sets Paul Blain Levy, University of Birmingham June 14, 2019 Abstract ZFC has sentences that quantify over all sets or all ordinals, without restriction. Some have argued that sen- tences of this kind lack a determinate meaning. We propose a set theory called TOPS, using Natural Deduction, that avoids this problem by speaking only about particular sets. 1 Introduction ZFC has long been established as a set-theoretic foundation of mathematics, but concerns about its meaningfulness have often been raised. Specifically, its use of unrestricted quantifiers seems to presuppose an absolute totality of sets. The purpose of this paper is to present a new set theory called TOPS that addresses this concern by speaking only about particular sets. Though weaker than ZFC, it is adequate for a large amount of mathematics. To explain the need for a new theory, we begin in Section 2 by surveying some basic mathematical beliefs and conceptions. Section 3 presents the language of TOPS, and Section 4 the theory itself, using Natural Deduction. Section 5 adapts the theory to allow theorems with free variables. Related work is discussed in Section 6. While TOPS is by no means the first set theory to use restricted quantifiers in order to avoid assuming a totality of sets, it is the first to do so using Natural Deduction, which turns out to be very helpful. This is apparent when TOPS is compared to a previously studied system [26] that proves essentially the same sentences but includes an inference rule that (we argue) cannot be considered truth-preserving. -
Real Valued Functions
REAL VALUED FUNCTIONS In mathematics, a real-valued function is a function whose domain is a subset D R of the set R of real numbers and the codomain is R; such a function can be represented by a graph in the Cartesian plane. In simplest terms the domain of a function is the set of all values that can be plugged into a function and have the function exist and have a real number for a value. So, for the domain we need to avoid division by zero, square roots of negative numbers, logarithms of zero and logarithms of negative numbers. The range of a function is simply the set of all possible values that a function can take. Continuous functions. A function f is said to be continuous if, roughly speaking, its graph can be drawn without lifting the pen so that such a graph is a curve with no "holes" or "jumps" or “breaks”. Intuitively, that’s easy to understand but we need to make this definition mathematically rigorous. Definition in terms of limits of functions Assume that is a function defined on an interval The function f is continuous at some point c of its domain if the limit of f(x) as x approaches c exists and is equal to f(c). In mathematical notation, this is written as In detail this means three conditions: a) first, f has to be defined at c. b) second, the limit on the left hand side of that equation has to exist. c) third, the value of this limit must equal f(c). -
APA Eastern Division 2019 Annual Meeting Program
The American Philosophical Association EASTERN DIVISION ONE HUNDRED FIFTEENTH ANNUAL MEETING PROGRAM SHERATON NEW YORK TIMES SQUARE NEW YORK, NEW YORK JANUARY 7 – 10, 2019 Visit our table at APA Eastern OFFERING A 20% (PB) / 40% (HC) DISCOUNT WITH FREE SHIPPING TO THE CONTIGUOUS U.S. FOR ORDERS PLACED AT THE CONFERENCE. THE POETRY OF APPROACHING HEGEL’S LOGIC, GEORGES BATAILLE OBLIQUELY Georges Bataille Melville, Molière, Beckett Translated and with an Introduction by Angelica Nuzzo Stuart Kendall THE POLITICS OF PARADIGMS ZHUANGZI AND THE Thomas S. Kuhn, James B. Conant, BECOMING OF NOTHINGNESS and the Cold War “Struggle for David Chai Men’s Minds” George A. Reisch ANOTHER AVAILABLE APRIL 2019 WHITE MAN’S BURDEN Josiah Royce’s Quest for a Philosophy THE REAL METAPHYSICAL CLUB of white Racial Empire The Philosophers, Their Debates, and Tommy J. Curry Selected Writings from 1870 to 1885 Frank X. Ryan, Brian E. Butler, and BOUNDARY LINES James A. Good, editors Philosophy and Postcolonialism Introduction by John R. Shook Emanuela Fornari AVAILABLE MARCH 2019 Translated by Iain Halliday Foreword by Étienne Balibar PRAGMATISM APPLIED William James and the Challenges THE CUDGEL AND THE CARESS of Contemporary Life Reflections on Cruelty and Tenderness Clifford S. Stagoll and David Farrell Krell Michael P. Levine, editors AVAILABLE MARCH 2019 AVAILABLE APRIL 2019 LOVE AND VIOLENCE BUDDHIST FEMINISMS The Vexatious Factors of Civilization AND FEMININITIES Lea Melandri Karma Lekshe Tsomo, editor Translated by Antonio Calcagno www.sunypress.edu II IMPORTANT NOTICES FOR MEETING ATTENDEES SESSION LOCATIONS Please note: this online version of the program does not include session locations. -
SYLLABUS Gödel's Theorems Phil 93902 TR 3:30 143 Debartolo Hall
SYLLABUS G¨odel's Theorems Phil 93902 TR 3:30 143 Debartolo Hall Michael Detlefsen ([email protected]), 408 Malloy Hall Curtis Franks ([email protected]), 411 Malloy Hall Fall 2007 I. Course Objectives • To acquaint the student with the basic ideas, methods and techniques behind one of the most striking scientific achievements of the twentieth century, the discovery of G¨odel'sincompleteness theorems. • To engage the student in productive thinking concerning the broader significance of these theorems. II. Texts • Kurt G¨odel:Collected Works, vol. I, Feferman et al. (eds.), Oxford, Oxford U. Press, 1986. III. Organization and Projected Schedule The course will divided into the following units, each estimated to run the indicated number of meetings. We plan to cover all of units A through G. We will cover the additional units according to the interests of the seminar participants. A. Historical and conceptual background and basic terminology. Two meetings. B. Basic mathematical background. Proof of the first theorem (G1). Four to five meet- ings. 1 C. Modifications of the proof of the first theorem, esp. Rosser's modification. One to two meetings. D. Proof of the second theorem (G2). Central task is to clarify the proofs of the second and third of the Hilbert-Bernays-L¨obDerivability Conditions (HBL Conditions). Two meetings. E. Alternative proofs of G2. Jeroslow's 1973 proof. Feferman's 1960 proof. Freidman's recent proof. [Note: We may discuss Jeroslow's and Feferman's proofs in inverse historical order because Jeroslow's proof is in some ways the less radical departure from the Hilbert-Bernays-L¨obproof.] Three meetings.