Unions and the Axiom of Choice
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Mathematics 144 Set Theory Fall 2012 Version
MATHEMATICS 144 SET THEORY FALL 2012 VERSION Table of Contents I. General considerations.……………………………………………………………………………………………………….1 1. Overview of the course…………………………………………………………………………………………………1 2. Historical background and motivation………………………………………………………….………………4 3. Selected problems………………………………………………………………………………………………………13 I I. Basic concepts. ………………………………………………………………………………………………………………….15 1. Topics from logic…………………………………………………………………………………………………………16 2. Notation and first steps………………………………………………………………………………………………26 3. Simple examples…………………………………………………………………………………………………………30 I I I. Constructions in set theory.………………………………………………………………………………..……….34 1. Boolean algebra operations.……………………………………………………………………………………….34 2. Ordered pairs and Cartesian products……………………………………………………………………… ….40 3. Larger constructions………………………………………………………………………………………………..….42 4. A convenient assumption………………………………………………………………………………………… ….45 I V. Relations and functions ……………………………………………………………………………………………….49 1.Binary relations………………………………………………………………………………………………………… ….49 2. Partial and linear orderings……………………………..………………………………………………… ………… 56 3. Functions…………………………………………………………………………………………………………… ….…….. 61 4. Composite and inverse function.…………………………………………………………………………… …….. 70 5. Constructions involving functions ………………………………………………………………………… ……… 77 6. Order types……………………………………………………………………………………………………… …………… 80 i V. Number systems and set theory …………………………………………………………………………………. 84 1. The Natural Numbers and Integers…………………………………………………………………………….83 2. Finite induction -
Algebra I Chapter 1. Basic Facts from Set Theory 1.1 Glossary of Abbreviations
Notes: c F.P. Greenleaf, 2000-2014 v43-s14sets.tex (version 1/1/2014) Algebra I Chapter 1. Basic Facts from Set Theory 1.1 Glossary of abbreviations. Below we list some standard math symbols that will be used as shorthand abbreviations throughout this course. means “for all; for every” • ∀ means “there exists (at least one)” • ∃ ! means “there exists exactly one” • ∃ s.t. means “such that” • = means “implies” • ⇒ means “if and only if” • ⇐⇒ x A means “the point x belongs to a set A;” x / A means “x is not in A” • ∈ ∈ N denotes the set of natural numbers (counting numbers) 1, 2, 3, • · · · Z denotes the set of all integers (positive, negative or zero) • Q denotes the set of rational numbers • R denotes the set of real numbers • C denotes the set of complex numbers • x A : P (x) If A is a set, this denotes the subset of elements x in A such that •statement { ∈ P (x)} is true. As examples of the last notation for specifying subsets: x R : x2 +1 2 = ( , 1] [1, ) { ∈ ≥ } −∞ − ∪ ∞ x R : x2 +1=0 = { ∈ } ∅ z C : z2 +1=0 = +i, i where i = √ 1 { ∈ } { − } − 1.2 Basic facts from set theory. Next we review the basic definitions and notations of set theory, which will be used throughout our discussions of algebra. denotes the empty set, the set with nothing in it • ∅ x A means that the point x belongs to a set A, or that x is an element of A. • ∈ A B means A is a subset of B – i.e. -
COMPSCI 501: Formal Language Theory Insights on Computability Turing Machines Are a Model of Computation Two (No Longer) Surpris
Insights on Computability Turing machines are a model of computation COMPSCI 501: Formal Language Theory Lecture 11: Turing Machines Two (no longer) surprising facts: Marius Minea Although simple, can describe everything [email protected] a (real) computer can do. University of Massachusetts Amherst Although computers are powerful, not everything is computable! Plus: “play” / program with Turing machines! 13 February 2019 Why should we formally define computation? Must indeed an algorithm exist? Back to 1900: David Hilbert’s 23 open problems Increasingly a realization that sometimes this may not be the case. Tenth problem: “Occasionally it happens that we seek the solution under insufficient Given a Diophantine equation with any number of un- hypotheses or in an incorrect sense, and for this reason do not succeed. known quantities and with rational integral numerical The problem then arises: to show the impossibility of the solution under coefficients: To devise a process according to which the given hypotheses or in the sense contemplated.” it can be determined in a finite number of operations Hilbert, 1900 whether the equation is solvable in rational integers. This asks, in effect, for an algorithm. Hilbert’s Entscheidungsproblem (1928): Is there an algorithm that And “to devise” suggests there should be one. decides whether a statement in first-order logic is valid? Church and Turing A Turing machine, informally Church and Turing both showed in 1936 that a solution to the Entscheidungsproblem is impossible for the theory of arithmetic. control To make and prove such a statement, one needs to define computability. In a recent paper Alonzo Church has introduced an idea of “effective calculability”, read/write head which is equivalent to my “computability”, but is very differently defined. -
A Class of Symmetric Difference-Closed Sets Related to Commuting Involutions John Campbell
A class of symmetric difference-closed sets related to commuting involutions John Campbell To cite this version: John Campbell. A class of symmetric difference-closed sets related to commuting involutions. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2017, Vol 19 no. 1. hal-01345066v4 HAL Id: hal-01345066 https://hal.archives-ouvertes.fr/hal-01345066v4 Submitted on 18 Mar 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Discrete Mathematics and Theoretical Computer Science DMTCS vol. 19:1, 2017, #8 A class of symmetric difference-closed sets related to commuting involutions John M. Campbell York University, Canada received 19th July 2016, revised 15th Dec. 2016, 1st Feb. 2017, accepted 10th Feb. 2017. Recent research on the combinatorics of finite sets has explored the structure of symmetric difference-closed sets, and recent research in combinatorial group theory has concerned the enumeration of commuting involutions in Sn and An. In this article, we consider an interesting combination of these two subjects, by introducing classes of symmetric difference-closed sets of elements which correspond in a natural way to commuting involutions in Sn and An. We consider the natural combinatorial problem of enumerating symmetric difference-closed sets consisting of subsets of sets consisting of pairwise disjoint 2-subsets of [n], and the problem of enumerating symmetric difference-closed sets consisting of elements which correspond to commuting involutions in An. -
MATH 361 Homework 9
MATH 361 Homework 9 Royden 3.3.9 First, we show that for any subset E of the real numbers, Ec + y = (E + y)c (translating the complement is equivalent to the complement of the translated set). Without loss of generality, assume E can be written as c an open interval (e1; e2), so that E + y is represented by the set fxjx 2 (−∞; e1 + y) [ (e2 + y; +1)g. This c is equal to the set fxjx2 = (e1 + y; e2 + y)g, which is equivalent to the set (E + y) . Second, Let B = A − y. From Homework 8, we know that outer measure is invariant under translations. Using this along with the fact that E is measurable: m∗(A) = m∗(B) = m∗(B \ E) + m∗(B \ Ec) = m∗((B \ E) + y) + m∗((B \ Ec) + y) = m∗(((A − y) \ E) + y) + m∗(((A − y) \ Ec) + y) = m∗(A \ (E + y)) + m∗(A \ (Ec + y)) = m∗(A \ (E + y)) + m∗(A \ (E + y)c) The last line follows from Ec + y = (E + y)c. Royden 3.3.10 First, since E1;E2 2 M and M is a σ-algebra, E1 [ E2;E1 \ E2 2 M. By the measurability of E1 and E2: ∗ ∗ ∗ c m (E1) = m (E1 \ E2) + m (E1 \ E2) ∗ ∗ ∗ c m (E2) = m (E2 \ E1) + m (E2 \ E1) ∗ ∗ ∗ ∗ c ∗ c m (E1) + m (E2) = 2m (E1 \ E2) + m (E1 \ E2) + m (E1 \ E2) ∗ ∗ ∗ c ∗ c = m (E1 \ E2) + [m (E1 \ E2) + m (E1 \ E2) + m (E1 \ E2)] c c Second, E1 \ E2, E1 \ E2, and E1 \ E2 are disjoint sets whose union is equal to E1 [ E2. -
Natural Numerosities of Sets of Tuples
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 367, Number 1, January 2015, Pages 275–292 S 0002-9947(2014)06136-9 Article electronically published on July 2, 2014 NATURAL NUMEROSITIES OF SETS OF TUPLES MARCO FORTI AND GIUSEPPE MORANA ROCCASALVO Abstract. We consider a notion of “numerosity” for sets of tuples of natural numbers that satisfies the five common notions of Euclid’s Elements, so it can agree with cardinality only for finite sets. By suitably axiomatizing such a no- tion, we show that, contrasting to cardinal arithmetic, the natural “Cantorian” definitions of order relation and arithmetical operations provide a very good algebraic structure. In fact, numerosities can be taken as the non-negative part of a discretely ordered ring, namely the quotient of a formal power series ring modulo a suitable (“gauge”) ideal. In particular, special numerosities, called “natural”, can be identified with the semiring of hypernatural numbers of appropriate ultrapowers of N. Introduction In this paper we consider a notion of “equinumerosity” on sets of tuples of natural numbers, i.e. an equivalence relation, finer than equicardinality, that satisfies the celebrated five Euclidean common notions about magnitudines ([8]), including the principle that “the whole is greater than the part”. This notion preserves the basic properties of equipotency for finite sets. In particular, the natural Cantorian definitions endow the corresponding numerosities with a structure of a discretely ordered semiring, where 0 is the size of the emptyset, 1 is the size of every singleton, and greater numerosity corresponds to supersets. This idea of “Euclidean” numerosity has been recently investigated by V. -
Families of Sets and Extended Operations Families of Sets
Families of Sets and Extended Operations Families of Sets When dealing with sets whose elements are themselves sets it is fairly common practice to refer to them as families of sets, however this is not a definition. In fact, technically, a family of sets need not be a set, because we allow repeated elements, so a family is a multiset. However, we do require that when repeated elements appear they are distinguishable. F = {A , A , A , A } with A = {a,b,c}, A ={a}, A = {a,d} and 1 2 3 4 1 2 3 A = {a} is a family of sets. 4 Extended Union and Intersection Let F be a family of sets. Then we define: The union over F by: ∪ A={x :∃ A∈F x∈A}= {x :∃ A A∈F∧x∈A} A∈F and the intersection over F by: ∩ A = {x :∀ A∈F x∈A}= {x :∀ A A∈F ⇒ x∈A}. A∈F For example, with F = {A , A , A , A } where A = {a,b,c}, 1 2 3 4 1 A ={a}, A = {a,d} and A = {a} we have: 2 3 4 ∪ A = {a ,b , c , d } and ∩ A = {a}. A∈F A∈F Theorem 2.8 For each set B in a family F of sets, a) ∩ A ⊆ B A∈F b) B ⊆ ∪ A. A∈F Pf: a) Suppose x ∈ ∩ A, then ∀A ∈ F, x ∈ A. Since B ∈ F, we have x ∈ B. Thus, ∩ A ⊆ B. b) Now suppose y ∈ B. Since B ∈ F, y ∈ ∪ A. Thus, B ⊆ ∪ A. Caveat Care must be taken with the empty family F, i.e., the family containing no sets. -
Learning from Streams
Learning from Streams Sanjay Jain?1, Frank Stephan??2 and Nan Ye3 1 Department of Computer Science, National University of Singapore, Singapore 117417, Republic of Singapore. [email protected] 2 Department of Computer Science and Department of Mathematics, National University of Singapore, Singapore 117417, Republic of Singapore. [email protected] 3 Department of Computer Science, National University of Singapore, Singapore 117417, Republic of Singapore. [email protected] Abstract. Learning from streams is a process in which a group of learn- ers separately obtain information about the target to be learned, but they can communicate with each other in order to learn the target. We are interested in machine models for learning from streams and study its learning power (as measured by the collection of learnable classes). We study how the power of learning from streams depends on the two parameters m and n, where n is the number of learners which track a single stream of input each and m is the number of learners (among the n learners) which have to find, in the limit, the right description of the target. We study for which combinations m, n and m0, n0 the following inclusion holds: Every class learnable from streams with parameters m, n is also learnable from streams with parameters m0, n0. For the learning of uniformly recursive classes, we get a full characterization which depends m only on the ratio n ; but for general classes the picture is more compli- cated. Most of the noninclusions in team learning carry over to nonin- clusions with the same parameters in the case of learning from streams; but only few inclusions are preserved and some additional noninclusions hold. -
Primitive Recursive Functions Are Recursively Enumerable
AN ENUMERATION OF THE PRIMITIVE RECURSIVE FUNCTIONS WITHOUT REPETITION SHIH-CHAO LIU (Received April 15,1900) In a theorem and its corollary [1] Friedberg gave an enumeration of all the recursively enumerable sets without repetition and an enumeration of all the partial recursive functions without repetition. This note is to prove a similar theorem for the primitive recursive functions. The proof is only a classical one. We shall show that the theorem is intuitionistically unprovable in the sense of Kleene [2]. For similar reason the theorem by Friedberg is also intuitionistical- ly unprovable, which is not stated in his paper. THEOREM. There is a general recursive function ψ(n, a) such that the sequence ψ(0, a), ψ(l, α), is an enumeration of all the primitive recursive functions of one variable without repetition. PROOF. Let φ(n9 a) be an enumerating function of all the primitive recursive functions of one variable, (See [3].) We define a general recursive function v(a) as follows. v(0) = 0, v(n + 1) = μy, where μy is the least y such that for each j < n + 1, φ(y, a) =[= φ(v(j), a) for some a < n + 1. It is noted that the value v(n + 1) can be found by a constructive method, for obviously there exists some number y such that the primitive recursive function <p(y> a) takes a value greater than all the numbers φ(v(0), 0), φ(y(Ϋ), 0), , φ(v(n\ 0) f or a = 0 Put ψ(n, a) = φ{v{n), a). -
Solutions to Homework 2 Math 55B
Solutions to Homework 2 Math 55b 1. Give an example of a subset A ⊂ R such that, by repeatedly taking closures and complements. Take A := (−5; −4)−Q [[−4; −3][f2g[[−1; 0][(1; 2)[(2; 3)[ (4; 5)\Q . (The number of intervals in this sets is unnecessarily large, but, as Kevin suggested, taking a set that is complement-symmetric about 0 cuts down the verification in half. Note that this set is the union of f0g[(1; 2)[(2; 3)[ ([4; 5] \ Q) and the complement of the reflection of this set about the the origin). Because of this symmetry, we only need to verify that the closure- complement-closure sequence of the set f0g [ (1; 2) [ (2; 3) [ (4; 5) \ Q consists of seven distinct sets. Here are the (first) seven members of this sequence; from the eighth member the sets start repeating period- ically: f0g [ (1; 2) [ (2; 3) [ (4; 5) \ Q ; f0g [ [1; 3] [ [4; 5]; (−∞; 0) [ (0; 1) [ (3; 4) [ (5; 1); (−∞; 1] [ [3; 4] [ [5; 1); (1; 3) [ (4; 5); [1; 3] [ [4; 5]; (−∞; 1) [ (3; 4) [ (5; 1). Thus, by symmetry, both the closure- complement-closure and complement-closure-complement sequences of A consist of seven distinct sets, and hence there is a total of 14 different sets obtained from A by taking closures and complements. Remark. That 14 is the maximal possible number of sets obtainable for any metric space is the Kuratowski complement-closure problem. This is proved by noting that, denoting respectively the closure and com- plement operators by a and b and by e the identity operator, the relations a2 = a; b2 = e, and aba = abababa take place, and then one can simply list all 14 elements of the monoid ha; b j a2 = a; b2 = e; aba = abababai. -
Polya Enumeration Theorem
Polya Enumeration Theorem Sasha Patotski Cornell University [email protected] December 11, 2015 Sasha Patotski (Cornell University) Polya Enumeration Theorem December 11, 2015 1 / 10 Cosets A left coset of H in G is gH where g 2 G (H is on the right). A right coset of H in G is Hg where g 2 G (H is on the left). Theorem If two left cosets of H in G intersect, then they coincide, and similarly for right cosets. Thus, G is a disjoint union of left cosets of H and also a disjoint union of right cosets of H. Corollary(Lagrange's theorem) If G is a finite group and H is a subgroup of G, then the order of H divides the order of G. In particular, the order of every element of G divides the order of G. Sasha Patotski (Cornell University) Polya Enumeration Theorem December 11, 2015 2 / 10 Applications of Lagrange's Theorem Theorem n! For any integers n ≥ 0 and 0 ≤ m ≤ n, the number m!(n−m)! is an integer. Theorem (ab)! (ab)! For any positive integers a; b the ratios (a!)b and (a!)bb! are integers. Theorem For an integer m > 1 let '(m) be the number of invertible numbers modulo m. For m ≥ 3 the number '(m) is even. Sasha Patotski (Cornell University) Polya Enumeration Theorem December 11, 2015 3 / 10 Polya's Enumeration Theorem Theorem Suppose that a finite group G acts on a finite set X . Then the number of colorings of X in n colors inequivalent under the action of G is 1 X N(n) = nc(g) jGj g2G where c(g) is the number of cycles of g as a permutation of X . -
MTH 304: General Topology Semester 2, 2017-2018
MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds..........................