Examples of Proof: Sets We Discussed in Class How to Formally Show That One Set Is a Subset of Another and How to Show Two Sets Are Equal
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Reasoning About Equations Using Equality
Mathematics Instructional Plan – Grade 4 Reasoning About Equations Using Equality Strand: Patterns, Functions, Algebra Topic: Using reasoning to assess the equality in closed and open arithmetic equations Primary SOL: 4.16 The student will recognize and demonstrate the meaning of equality in an equation. Related SOL: 4.4a Materials: Partners Explore Equality and Equations activity sheet (attached) Vocabulary equal symbol (=), equality, equation, expression, not equal symbol (≠), number sentence, open number sentence, reasoning Student/Teacher Actions: What should students be doing? What should teachers be doing? 1. Activate students’ prior knowledge about the equal sign. Formally assess what students already know and what misconceptions they have. Ask students to open their notebooks to a clean sheet of paper and draw lines to divide the sheet into thirds as shown below. Create a poster of the same graphic organizer for the room. This activity should take no more than 10 minutes; it is not a teaching opportunity but simply a data- collection activity. Equality and the Equal Sign (=) I know that- I wonder if- I learned that- 1 1 1 a. Let students know they will be creating a graphic organizer about equality and the equal sign; that is, what it means and how it is used. The organizer is to help them keep track of what they know, what they wonder, and what they learn. The “I know that” column is where they can write things they are pretty sure are correct. The “I wonder if” column is where they can write things they have questions about and also where they can put things they think may be true but they are not sure. -
6.5 the Recursion Theorem
6.5. THE RECURSION THEOREM 417 6.5 The Recursion Theorem The recursion Theorem, due to Kleene, is a fundamental result in recursion theory. Theorem 6.5.1 (Recursion Theorem, Version 1 )Letϕ0,ϕ1,... be any ac- ceptable indexing of the partial recursive functions. For every total recursive function f, there is some n such that ϕn = ϕf(n). The recursion Theorem can be strengthened as follows. Theorem 6.5.2 (Recursion Theorem, Version 2 )Letϕ0,ϕ1,... be any ac- ceptable indexing of the partial recursive functions. There is a total recursive function h such that for all x ∈ N,ifϕx is total, then ϕϕx(h(x)) = ϕh(x). 418 CHAPTER 6. ELEMENTARY RECURSIVE FUNCTION THEORY A third version of the recursion Theorem is given below. Theorem 6.5.3 (Recursion Theorem, Version 3 ) For all n ≥ 1, there is a total recursive function h of n +1 arguments, such that for all x ∈ N,ifϕx is a total recursive function of n +1arguments, then ϕϕx(h(x,x1,...,xn),x1,...,xn) = ϕh(x,x1,...,xn), for all x1,...,xn ∈ N. As a first application of the recursion theorem, we can show that there is an index n such that ϕn is the constant function with output n. Loosely speaking, ϕn prints its own name. Let f be the recursive function such that f(x, y)=x for all x, y ∈ N. 6.5. THE RECURSION THEOREM 419 By the s-m-n Theorem, there is a recursive function g such that ϕg(x)(y)=f(x, y)=x for all x, y ∈ N. -
April 22 7.1 Recursion Theorem
CSE 431 Theory of Computation Spring 2014 Lecture 7: April 22 Lecturer: James R. Lee Scribe: Eric Lei Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. They may be distributed outside this class only with the permission of the Instructor. An interesting question about Turing machines is whether they can reproduce themselves. A Turing machine cannot be be defined in terms of itself, but can it still somehow print its own source code? The answer to this question is yes, as we will see in the recursion theorem. Afterward we will see some applications of this result. 7.1 Recursion Theorem Our end goal is to have a Turing machine that prints its own source code and operates on it. Last lecture we proved the existence of a Turing machine called SELF that ignores its input and prints its source code. We construct a similar proof for the recursion theorem. We will also need the following lemma proved last lecture. ∗ ∗ Lemma 7.1 There exists a computable function q :Σ ! Σ such that q(w) = hPwi, where Pw is a Turing machine that prints w and hats. Theorem 7.2 (Recursion theorem) Let T be a Turing machine that computes a function t :Σ∗ × Σ∗ ! Σ∗. There exists a Turing machine R that computes a function r :Σ∗ ! Σ∗, where for every w, r(w) = t(hRi; w): The theorem says that for an arbitrary computable function t, there is a Turing machine R that computes t on hRi and some input. Proof: We construct a Turing Machine R in three parts, A, B, and T , where T is given by the statement of the theorem. -
Intuitionism on Gödel's First Incompleteness Theorem
The Yale Undergraduate Research Journal Volume 2 Issue 1 Spring 2021 Article 31 2021 Incomplete? Or Indefinite? Intuitionism on Gödel’s First Incompleteness Theorem Quinn Crawford Yale University Follow this and additional works at: https://elischolar.library.yale.edu/yurj Part of the Mathematics Commons, and the Philosophy Commons Recommended Citation Crawford, Quinn (2021) "Incomplete? Or Indefinite? Intuitionism on Gödel’s First Incompleteness Theorem," The Yale Undergraduate Research Journal: Vol. 2 : Iss. 1 , Article 31. Available at: https://elischolar.library.yale.edu/yurj/vol2/iss1/31 This Article is brought to you for free and open access by EliScholar – A Digital Platform for Scholarly Publishing at Yale. It has been accepted for inclusion in The Yale Undergraduate Research Journal by an authorized editor of EliScholar – A Digital Platform for Scholarly Publishing at Yale. For more information, please contact [email protected]. Incomplete? Or Indefinite? Intuitionism on Gödel’s First Incompleteness Theorem Cover Page Footnote Written for Professor Sun-Joo Shin’s course PHIL 437: Philosophy of Mathematics. This article is available in The Yale Undergraduate Research Journal: https://elischolar.library.yale.edu/yurj/vol2/iss1/ 31 Crawford: Intuitionism on Gödel’s First Incompleteness Theorem Crawford | Philosophy Incomplete? Or Indefinite? Intuitionism on Gödel’s First Incompleteness Theorem By Quinn Crawford1 1Department of Philosophy, Yale University ABSTRACT This paper analyzes two natural-looking arguments that seek to leverage Gödel’s first incompleteness theo- rem for and against intuitionism, concluding in both cases that the argument is unsound because it equivo- cates on the meaning of “proof,” which differs between formalism and intuitionism. -
Chapter 1 Logic and Set Theory
Chapter 1 Logic and Set Theory To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. – Ian Stewart Does God play dice? The mathematics of chaos In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. That is, a proof is a logical argument, not an empir- ical one. One must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics. An unproven proposition for which there is some sort of empirical evidence is known as a conjecture. Mathematical logic is the framework upon which rigorous proofs are built. It is the study of the principles and criteria of valid inference and demonstrations. Logicians have analyzed set theory in great details, formulating a collection of axioms that affords a broad enough and strong enough foundation to mathematical reasoning. The standard form of axiomatic set theory is denoted ZFC and it consists of the Zermelo-Fraenkel (ZF) axioms combined with the axiom of choice (C). Each of the axioms included in this theory expresses a property of sets that is widely accepted by mathematicians. It is unfortunately true that careless use of set theory can lead to contradictions. Avoiding such contradictions was one of the original motivations for the axiomatization of set theory. 1 2 CHAPTER 1. LOGIC AND SET THEORY A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic. -
Theorem Proving in Classical Logic
MEng Individual Project Imperial College London Department of Computing Theorem Proving in Classical Logic Supervisor: Dr. Steffen van Bakel Author: David Davies Second Marker: Dr. Nicolas Wu June 16, 2021 Abstract It is well known that functional programming and logic are deeply intertwined. This has led to many systems capable of expressing both propositional and first order logic, that also operate as well-typed programs. What currently ties popular theorem provers together is their basis in intuitionistic logic, where one cannot prove the law of the excluded middle, ‘A A’ – that any proposition is either true or false. In classical logic this notion is provable, and the_: corresponding programs turn out to be those with control operators. In this report, we explore and expand upon the research about calculi that correspond with classical logic; and the problems that occur for those relating to first order logic. To see how these calculi behave in practice, we develop and implement functional languages for propositional and first order logic, expressing classical calculi in the setting of a theorem prover, much like Agda and Coq. In the first order language, users are able to define inductive data and record types; importantly, they are able to write computable programs that have a correspondence with classical propositions. Acknowledgements I would like to thank Steffen van Bakel, my supervisor, for his support throughout this project and helping find a topic of study based on my interests, for which I am incredibly grateful. His insight and advice have been invaluable. I would also like to thank my second marker, Nicolas Wu, for introducing me to the world of dependent types, and suggesting useful resources that have aided me greatly during this report. -
On the Mathematical Foundations of Learning
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 39, Number 1, Pages 1{49 S 0273-0979(01)00923-5 Article electronically published on October 5, 2001 ON THE MATHEMATICAL FOUNDATIONS OF LEARNING FELIPE CUCKER AND STEVE SMALE The problem of learning is arguably at the very core of the problem of intelligence, both biological and artificial. T. Poggio and C.R. Shelton Introduction (1) A main theme of this report is the relationship of approximation to learning and the primary role of sampling (inductive inference). We try to emphasize relations of the theory of learning to the mainstream of mathematics. In particular, there are large roles for probability theory, for algorithms such as least squares,andfor tools and ideas from linear algebra and linear analysis. An advantage of doing this is that communication is facilitated and the power of core mathematics is more easily brought to bear. We illustrate what we mean by learning theory by giving some instances. (a) The understanding of language acquisition by children or the emergence of languages in early human cultures. (b) In Manufacturing Engineering, the design of a new wave of machines is an- ticipated which uses sensors to sample properties of objects before, during, and after treatment. The information gathered from these samples is to be analyzed by the machine to decide how to better deal with new input objects (see [43]). (c) Pattern recognition of objects ranging from handwritten letters of the alpha- bet to pictures of animals, to the human voice. Understanding the laws of learning plays a large role in disciplines such as (Cog- nitive) Psychology, Animal Behavior, Economic Decision Making, all branches of Engineering, Computer Science, and especially the study of human thought pro- cesses (how the brain works). -
Plato on the Foundations of Modern Theorem Provers
The Mathematics Enthusiast Volume 13 Number 3 Number 3 Article 8 8-2016 Plato on the foundations of Modern Theorem Provers Ines Hipolito Follow this and additional works at: https://scholarworks.umt.edu/tme Part of the Mathematics Commons Let us know how access to this document benefits ou.y Recommended Citation Hipolito, Ines (2016) "Plato on the foundations of Modern Theorem Provers," The Mathematics Enthusiast: Vol. 13 : No. 3 , Article 8. Available at: https://scholarworks.umt.edu/tme/vol13/iss3/8 This Article is brought to you for free and open access by ScholarWorks at University of Montana. It has been accepted for inclusion in The Mathematics Enthusiast by an authorized editor of ScholarWorks at University of Montana. For more information, please contact [email protected]. TME, vol. 13, no.3, p.303 Plato on the foundations of Modern Theorem Provers Inês Hipolito1 Nova University of Lisbon Abstract: Is it possible to achieve such a proof that is independent of both acts and dispositions of the human mind? Plato is one of the great contributors to the foundations of mathematics. He discussed, 2400 years ago, the importance of clear and precise definitions as fundamental entities in mathematics, independent of the human mind. In the seventh book of his masterpiece, The Republic, Plato states “arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible or tangible objects into the argument” (525c). In the light of this thought, I will discuss the status of mathematical entities in the twentieth first century, an era when it is already possible to demonstrate theorems and construct formal axiomatic derivations of remarkable complexity with artificial intelligent agents the modern theorem provers. -
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. -
Part 3: First-Order Logic with Equality
Part 3: First-Order Logic with Equality Equality is the most important relation in mathematics and functional programming. In principle, problems in first-order logic with equality can be handled by, e.g., resolution theorem provers. Equality is theoretically difficult: First-order functional programming is Turing-complete. But: resolution theorem provers cannot even solve problems that are intuitively easy. Consequence: to handle equality efficiently, knowledge must be integrated into the theorem prover. 1 3.1 Handling Equality Naively Proposition 3.1: Let F be a closed first-order formula with equality. Let ∼ 2/ Π be a new predicate symbol. The set Eq(Σ) contains the formulas 8x (x ∼ x) 8x, y (x ∼ y ! y ∼ x) 8x, y, z (x ∼ y ^ y ∼ z ! x ∼ z) 8~x,~y (x1 ∼ y1 ^ · · · ^ xn ∼ yn ! f (x1, : : : , xn) ∼ f (y1, : : : , yn)) 8~x,~y (x1 ∼ y1 ^ · · · ^ xn ∼ yn ^ p(x1, : : : , xn) ! p(y1, : : : , yn)) for every f /n 2 Ω and p/n 2 Π. Let F~ be the formula that one obtains from F if every occurrence of ≈ is replaced by ∼. Then F is satisfiable if and only if Eq(Σ) [ fF~g is satisfiable. 2 Handling Equality Naively By giving the equality axioms explicitly, first-order problems with equality can in principle be solved by a standard resolution or tableaux prover. But this is unfortunately not efficient (mainly due to the transitivity and congruence axioms). 3 Roadmap How to proceed: • Arbitrary binary relations. • Equations (unit clauses with equality): Term rewrite systems. Expressing semantic consequence syntactically. Entailment for equations. • Equational clauses: Entailment for clauses with equality. -
Logic, Sets, and Proofs David A
Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Statements. A logical statement is a mathematical statement that is either true or false. Here we denote logical statements with capital letters A; B. Logical statements be combined to form new logical statements as follows: Name Notation Conjunction A and B Disjunction A or B Negation not A :A Implication A implies B if A, then B A ) B Equivalence A if and only if B A , B Here are some examples of conjunction, disjunction and negation: x > 1 and x < 3: This is true when x is in the open interval (1; 3). x > 1 or x < 3: This is true for all real numbers x. :(x > 1): This is the same as x ≤ 1. Here are two logical statements that are true: x > 4 ) x > 2. x2 = 1 , (x = 1 or x = −1). Note that \x = 1 or x = −1" is usually written x = ±1. Converses, Contrapositives, and Tautologies. We begin with converses and contrapositives: • The converse of \A implies B" is \B implies A". • The contrapositive of \A implies B" is \:B implies :A" Thus the statement \x > 4 ) x > 2" has: • Converse: x > 2 ) x > 4. • Contrapositive: x ≤ 2 ) x ≤ 4. 1 Some logical statements are guaranteed to always be true. These are tautologies. Here are two tautologies that involve converses and contrapositives: • (A if and only if B) , ((A implies B) and (B implies A)). In other words, A and B are equivalent exactly when both A ) B and its converse are true. -
Tool 1: Gender Terminology, Concepts and Definitions
TOOL 1 Gender terminology, concepts and def initions UNESCO Bangkok Office Asia and Pacific Regional Bureau for Education Mom Luang Pin Malakul Centenary Building 920 Sukhumvit Road, Prakanong, Klongtoei © Hadynyah/Getty Images Bangkok 10110, Thailand Email: [email protected] Website: https://bangkok.unesco.org Tel: +66-2-3910577 Fax: +66-2-3910866 Table of Contents Objectives .................................................................................................. 1 Key information: Setting the scene ............................................................. 1 Box 1: .......................................................................................................................2 Fundamental concepts: gender and sex ...................................................... 2 Box 2: .......................................................................................................................2 Self-study and/or group activity: Reflect on gender norms in your context........3 Self-study and/or group activity: Check understanding of definitions ................3 Gender and education ................................................................................. 4 Self-study and/or group activity: Gender and education definitions ...................4 Box 3: Gender parity index .....................................................................................7 Self-study and/or group activity: Reflect on girls’ and boys’ lives in your context .....................................................................................................................8