DOCSLIB.ORG

Bijections and Cardinality

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Bijections and Cardinality Bijections and Cardinality CS 2800: Discrete Structures, Spring 2015 Sid Chaudhuri Recap: Le t and Right !n"erses ● # function is injective $one-to-one) i' it has a left inverse – g : B → A is a left in"erse of f : A → B if g ( f (a) ) = a for all a ∈ A ● # function is surjective (onto) i' it has a right inverse – h : B → A is a right in"erse of f : A → B if f ( h (b) ) = b for all b ∈ B (hought for the Day #1 !s a left in"erse injecti"e or surjecti"e* +hy* !s a right in"erse injecti"e or surjecti"e* +hy* (Hint: how is f related to its left/right inverse?) Sur/injectivity o le t/right in"erses ● The left inverse is always surjective! – … since f is its right in"erse ● (he right in"erse is al-ays injecti"e. – / since f is its le t in"erse 0actoid for the Day #1 !f a function has both a left in"erse and a right in"erse, then the t-o in"erses are identical, and this com2on in"erse is unique (Prove!) (his is called the two-sided inverse, or usually just the inverse f –1 of the function f http:/,---.cs.cornell.edu,courses,cs2800,2015sp,handouts,jonpa56function_notes.pdf Bijection and two-sided in"erse ● # function f is bijecti"e i' it has a t-o-sided in"erse ● Proof (⇒): !f it is bijecti"e, it has a left in"erse $since injecti"e) and a right in"erse (since surjecti"e), which must be one and the sa2e by the previous factoid ● Proof (⇐): !f it has a t-o-sided in"erse, it is both injecti"e (since there is a left in"erse) and surjecti"e (since there is a right in"erse). 7ence it is bijecti"e. !n"erse o a unction ● (he in"erse of a bijecti"e function f : A → B is the unique function f ‑1: B → A such that for any a ∈ A, f ‑1(f(a)) = a and for any b ∈ B, f(f ‑1(b)) = b ● # function is bijecti"e i' it has an in"erse function f(a) a f ‑1(a) b = f(a) f A B f ‑1 0ollo-ing 8rnie Croot's slides !n"erse o a unction ● !f f is not a bijection, it cannot ha"e an in"erse function f x x 1 1 y ? y 2 2 z z 3 3 w w :nto, not one-to-one f -1(2) = ? 0ollo-ing 8rnie Croot's slides !n"erse o a unction ● !f f is not a bijection, it cannot ha"e an in"erse function f 1 x 1 x 2 2 y y 3 3 z ? z 4 4 :ne-to-one, not onto f -1(4) = ? 0ollo-ing 8rnie Croot's slides 7ow can we count elements in a set* 7ow can we count elements in a set* ● 8asy for finite sets – just count the elements. ● Does it e"en ma5e sense to ask a1out the num1er of elements in an infinite set* ● !s it meaningful to say one infinite set is larger than another* – #re the natural numbers larger than ● the e"en numbers* ● the rational numbers* ● the real numbers* 0ollo-ing 8rnie Croot's slides Cardinality and Bijections ● !f A and B are finite sets, clearly they ha"e the sa2e num1er of elements i' there is a bijection 1et-een them e.g. | {x, y, z} | = | {1, 2, 3} | = 3 x 1 y 2 z 3 0ollo-ing 8rnie Croot's slides Cardinality and Bijections ● Definition: Set A has the sa2e cardinality as set B, denoted |A| = |B|, i' there is a bijection from A to B – 0or finite sets, cardinality is the number of elements – (here is a bijection from n%element set A to {1, 2, 3, …, n } 0ollo-ing 8rnie Croot's slides Cardinality and Bijections ● =atural num1ers and e"en num1ers ha"e the sa2e cardinality 0 1 2 @ > ... 2 > ? 8 10 ... Sets having the same cardinality as the natural numbers (or some subset of the natural numbers) are called countable sets 0ollo-ing 8rnie Croot's slides Cardinality and Bijections ● =atural num1ers and rational num1ers ha"e the sa2e cardinality. 0 1 4 5 1 1 1 1 ⋯ 1 2 3 4 6 Illustrating proof 2 2 2 2 only for positive 1 2 3 rationals here, can 3 7 be easily extended 3 3 3 to all rationals 1 2 3 8 4 ⋱ 1 ⋮ Cardinality and Bijections ● (he natural num1ers and real num1ers do not ha"e the sa2e cardinality x1 0 . 0 0 0 0 0 0 0 0 0 … x2 0 . 1 0 3 0 4 0 5 0 1 … x3 0 . 9 8 7 6 5 4 3 2 1 … x4 0 . 0 1 2 1 2 1 2 1 2 … x5 ⁞ Cardinality and Bijections ● (he natural num1ers and real num1ers do not ha"e the sa2e cardinality Consider the number x 0 . 0 0 0 0 0 0 0 0 0 … 1 y = 0 . b1 b2 b3... x2 0 . 1 0 3 0 4 0 5 0 1 … 1 if the ith decimal place of x is zero b = i x3 0 . 9 8 7 6 5 4 3 2 1 … i 0 if it is non-zero x4 0 . 0 1 2 1 2 1 2 1 2 … y x x5 ⁞ cannot be equal to any i – it difers by one digit from each one. (here are many infinities (hought or the Day )2 Do the real inter"al [0, 1] and the unit square [0, 1] B [0, 1] ha"e the sa2e cardinality* 0,1 1,1 0 1 0,0 1,0 Comparing Cardinalities ● Definition: !f there is an injecti"e function from set A to set B, we say |A| ≤ |B| 2 > ? 8 ... 0 1 2 @ > 5 ? C 8 D ... |Evens| ≤ |N| 0ollo-ing 8rnie Croot's slides Comparing Cardinalities ● Definition: !f there is an injecti"e function from set A to set B, but not from B to A, we say |A| < |B| ● Cantor<SchrEder<Bernstein theorem: !f |A| ≤ |B| and |B| ≤ |A|, then |A| = |B| – Exercise: pro"e this. – i.e. sho- that there is a bijection from A to B if there are injective functions from A to B and from B to A – $it's not easy.& 0ollo-ing 8rnie Croot's slides.
Load more

Recommended publications
  • 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).
    [Show full text]
  • The Matroid Theorem We First Review Our Definitions: a Subset System Is A
    CMPSCI611: The Matroid Theorem Lecture 5 We first review our definitions: A subset system is a set E together with a set of subsets of E, called I, such that I is closed under inclusion. This means that if X ⊆ Y and Y ∈ I, then X ∈ I. The optimization problem for a subset system (E, I) has as input a positive weight for each element of E. Its output is a set X ∈ I such that X has at least as much total weight as any other set in I. A subset system is a matroid if it satisfies the exchange property: If i and i0 are sets in I and i has fewer elements than i0, then there exists an element e ∈ i0 \ i such that i ∪ {e} ∈ I. 1 The Generic Greedy Algorithm Given any finite subset system (E, I), we find a set in I as follows: • Set X to ∅. • Sort the elements of E by weight, heaviest first. • For each element of E in this order, add it to X iff the result is in I. • Return X. Today we prove: Theorem: For any subset system (E, I), the greedy al- gorithm solves the optimization problem for (E, I) if and only if (E, I) is a matroid. 2 Theorem: For any subset system (E, I), the greedy al- gorithm solves the optimization problem for (E, I) if and only if (E, I) is a matroid. Proof: We will show first that if (E, I) is a matroid, then the greedy algorithm is correct. Assume that (E, I) satisfies the exchange property.
    [Show full text]
  • Is Uncountable the Open Interval (0, 1)
    Section 2.4:R is uncountable (0, 1) is uncountable This assertion and its proof date back to the 1890’s and to Georg Cantor. The proof is often referred to as “Cantor’s diagonal argument” and applies in more general contexts than we will see in these notes. Our goal in this section is to show that the setR of real numbers is uncountable or non-denumerable; this means that its elements cannot be listed, or cannot be put in bijective correspondence with the natural numbers. We saw at the end of Section 2.3 that R has the same cardinality as the interval ( π , π ), or the interval ( 1, 1), or the interval (0, 1). We will − 2 2 − show that the open interval (0, 1) is uncountable. Georg Cantor : born in St Petersburg (1845), died in Halle (1918) Theorem 42 The open interval (0, 1) is not a countable set. Dr Rachel Quinlan MA180/MA186/MA190 Calculus R is uncountable 143 / 222 Dr Rachel Quinlan MA180/MA186/MA190 Calculus R is uncountable 144 / 222 The open interval (0, 1) is not a countable set A hypothetical bijective correspondence Our goal is to show that the interval (0, 1) cannot be put in bijective correspondence with the setN of natural numbers. Our strategy is to We recall precisely what this set is. show that no attempt at constructing a bijective correspondence between It consists of all real numbers that are greater than zero and less these two sets can ever be complete; it can never involve all the real than 1, or equivalently of all the points on the number line that are numbers in the interval (0, 1) no matter how it is devised.
    [Show full text]
  • Inverse of Exponential Functions Are Logarithmic Functions
    Math Instructional Framework Full Name Math III Unit 3 Lesson 2 Time Frame Unit Name Logarithmic Functions as Inverses of Exponential Functions Learning Task/Topics/ Task 2: How long Does It Take? Themes Task 3: The Population of Exponentia Task 4: Modeling Natural Phenomena on Earth Culminating Task: Traveling to Exponentia Standards and Elements MM3A2. Students will explore logarithmic functions as inverses of exponential functions. c. Define logarithmic functions as inverses of exponential functions. Lesson Essential Questions How can you graph the inverse of an exponential function? Activator PROBLEM 2.Task 3: The Population of Exponentia (Problem 1 could be completed prior) Work Session Inverse of Exponential Functions are Logarithmic Functions A Graph the inverse of exponential functions. B Graph logarithmic functions. See Notes Below. VOCABULARY Asymptote: A line or curve that describes the end behavior of the graph. A graph never crosses a vertical asymptote but it may cross a horizontal or oblique asymptote. Common logarithm: A logarithm with a base of 10. A common logarithm is the power, a, such that 10a = b. The common logarithm of x is written log x. For example, log 100 = 2 because 102 = 100. Exponential functions: A function of the form y = a·bx where a > 0 and either 0 < b < 1 or b > 1. Logarithmic functions: A function of the form y = logbx, with b 1 and b and x both positive. A logarithmic x function is the inverse of an exponential function. The inverse of y = b is y = logbx. Logarithm: The logarithm base b of a number x, logbx, is the power to which b must be raised to equal x.
    [Show full text]
  • Analysis of Functions of a Single Variable a Detailed Development
    ANALYSIS OF FUNCTIONS OF A SINGLE VARIABLE A DETAILED DEVELOPMENT LAWRENCE W. BAGGETT University of Colorado OCTOBER 29, 2006 2 For Christy My Light i PREFACE I have written this book primarily for serious and talented mathematics scholars , seniors or first-year graduate students, who by the time they finish their schooling should have had the opportunity to study in some detail the great discoveries of our subject. What did we know and how and when did we know it? I hope this book is useful toward that goal, especially when it comes to the great achievements of that part of mathematics known as analysis. I have tried to write a complete and thorough account of the elementary theories of functions of a single real variable and functions of a single complex variable. Separating these two subjects does not at all jive with their development historically, and to me it seems unnecessary and potentially confusing to do so. On the other hand, functions of several variables seems to me to be a very different kettle of fish, so I have decided to limit this book by concentrating on one variable at a time. Everyone is taught (told) in school that the area of a circle is given by the formula A = πr2: We are also told that the product of two negatives is a positive, that you cant trisect an angle, and that the square root of 2 is irrational. Students of natural sciences learn that eiπ = 1 and that sin2 + cos2 = 1: More sophisticated students are taught the Fundamental− Theorem of calculus and the Fundamental Theorem of Algebra.
    [Show full text]
  • IVC Factsheet Functions Comp Inverse
    Imperial Valley College Math Lab Functions: Composition and Inverse Functions FUNCTION COMPOSITION In order to perform a composition of functions, it is essential to be familiar with function notation. If you see something of the form “푓(푥) = [expression in terms of x]”, this means that whatever you see in the parentheses following f should be substituted for x in the expression. This can include numbers, variables, other expressions, and even other functions. EXAMPLE: 푓(푥) = 4푥2 − 13푥 푓(2) = 4 ∙ 22 − 13(2) 푓(−9) = 4(−9)2 − 13(−9) 푓(푎) = 4푎2 − 13푎 푓(푐3) = 4(푐3)2 − 13푐3 푓(ℎ + 5) = 4(ℎ + 5)2 − 13(ℎ + 5) Etc. A composition of functions occurs when one function is “plugged into” another function. The notation (푓 ○푔)(푥) is pronounced “푓 of 푔 of 푥”, and it literally means 푓(푔(푥)). In other words, you “plug” the 푔(푥) function into the 푓(푥) function. Similarly, (푔 ○푓)(푥) is pronounced “푔 of 푓 of 푥”, and it literally means 푔(푓(푥)). In other words, you “plug” the 푓(푥) function into the 푔(푥) function. WARNING: Be careful not to confuse (푓 ○푔)(푥) with (푓 ∙ 푔)(푥), which means 푓(푥) ∙ 푔(푥) . EXAMPLES: 푓(푥) = 4푥2 − 13푥 푔(푥) = 2푥 + 1 a. (푓 ○푔)(푥) = 푓(푔(푥)) = 4[푔(푥)]2 − 13 ∙ 푔(푥) = 4(2푥 + 1)2 − 13(2푥 + 1) = [푠푚푝푙푓푦] … = 16푥2 − 10푥 − 9 b. (푔 ○푓)(푥) = 푔(푓(푥)) = 2 ∙ 푓(푥) + 1 = 2(4푥2 − 13푥) + 1 = 8푥2 − 26푥 + 1 A function can even be “composed” with itself: c.
    [Show full text]
  • Determinacy in Linear Rational Expectations Models
    Journal of Mathematical Economics 40 (2004) 815–830 Determinacy in linear rational expectations models Stéphane Gauthier∗ CREST, Laboratoire de Macroéconomie (Timbre J-360), 15 bd Gabriel Péri, 92245 Malakoff Cedex, France Received 15 February 2002; received in revised form 5 June 2003; accepted 17 July 2003 Available online 21 January 2004 Abstract The purpose of this paper is to assess the relevance of rational expectations solutions to the class of linear univariate models where both the number of leads in expectations and the number of lags in predetermined variables are arbitrary. It recommends to rule out all the solutions that would fail to be locally unique, or equivalently, locally determinate. So far, this determinacy criterion has been applied to particular solutions, in general some steady state or periodic cycle. However solutions to linear models with rational expectations typically do not conform to such simple dynamic patterns but express instead the current state of the economic system as a linear difference equation of lagged states. The innovation of this paper is to apply the determinacy criterion to the sets of coefficients of these linear difference equations. Its main result shows that only one set of such coefficients, or the corresponding solution, is locally determinate. This solution is commonly referred to as the fundamental one in the literature. In particular, in the saddle point configuration, it coincides with the saddle stable (pure forward) equilibrium trajectory. © 2004 Published by Elsevier B.V. JEL classification: C32; E32 Keywords: Rational expectations; Selection; Determinacy; Saddle point property 1. Introduction The rational expectations hypothesis is commonly justified by the fact that individual forecasts are based on the relevant theory of the economic system.
    [Show full text]
  • SHEET 14: LINEAR ALGEBRA 14.1 Vector Spaces
    SHEET 14: LINEAR ALGEBRA Throughout this sheet, let F be a field. In examples, you need only consider the field F = R. 14.1 Vector spaces Definition 14.1. A vector space over F is a set V with two operations, V × V ! V :(x; y) 7! x + y (vector addition) and F × V ! V :(λ, x) 7! λx (scalar multiplication); that satisfy the following axioms. 1. Addition is commutative: x + y = y + x for all x; y 2 V . 2. Addition is associative: x + (y + z) = (x + y) + z for all x; y; z 2 V . 3. There is an additive identity 0 2 V satisfying x + 0 = x for all x 2 V . 4. For each x 2 V , there is an additive inverse −x 2 V satisfying x + (−x) = 0. 5. Scalar multiplication by 1 fixes vectors: 1x = x for all x 2 V . 6. Scalar multiplication is compatible with F :(λµ)x = λ(µx) for all λ, µ 2 F and x 2 V . 7. Scalar multiplication distributes over vector addition and over scalar addition: λ(x + y) = λx + λy and (λ + µ)x = λx + µx for all λ, µ 2 F and x; y 2 V . In this context, elements of F are called scalars and elements of V are called vectors. Definition 14.2. Let n be a nonnegative integer. The coordinate space F n = F × · · · × F is the set of all n-tuples of elements of F , conventionally regarded as column vectors. Addition and scalar multiplication are defined componentwise; that is, 2 3 2 3 2 3 2 3 x1 y1 x1 + y1 λx1 6x 7 6y 7 6x + y 7 6λx 7 6 27 6 27 6 2 2 7 6 27 if x = 6 .
    [Show full text]
  • PROBLEM SET 1. the AXIOM of FOUNDATION Early on in the Book
    PROBLEM SET 1. THE AXIOM OF FOUNDATION Early on in the book (page 6) it is indicated that throughout the formal development ‘set’ is going to mean ‘pure set’, or set whose elements, elements of elements, and so on, are all sets and not items of any other kind such as chairs or tables. This convention applies also to these problems. 1. A set y is called an epsilon-minimal element of a set x if y Î x, but there is no z Î x such that z Î y, or equivalently x Ç y = Ø. The axiom of foundation, also called the axiom of regularity, asserts that any set that has any element at all (any nonempty set) has an epsilon-minimal element. Show that this axiom implies the following: (a) There is no set x such that x Î x. (b) There are no sets x and y such that x Î y and y Î x. (c) There are no sets x and y and z such that x Î y and y Î z and z Î x. 2. In the book the axiom of foundation or regularity is considered only in a late chapter, and until that point no use of it is made of it in proofs. But some results earlier in the book become significantly easier to prove if one does use it. Show, for example, how to use it to give an easy proof of the existence for any sets x and y of a set x* such that x* and y are disjoint (have empty intersection) and there is a bijection (one-to-one onto function) from x to x*, a result called the exchange principle.
    [Show full text]
  • Two Fundamental Theorems About the Definite Integral
    Two Fundamental Theorems about the Definite Integral These lecture notes develop the theorem Stewart calls The Fundamental Theorem of Calculus in section 5.3. The approach I use is slightly different than that used by Stewart, but is based on the same fundamental ideas. 1 The definite integral Recall that the expression b f(x) dx ∫a is called the definite integral of f(x) over the interval [a,b] and stands for the area underneath the curve y = f(x) over the interval [a,b] (with the understanding that areas above the x-axis are considered positive and the areas beneath the axis are considered negative). In today's lecture I am going to prove an important connection between the definite integral and the derivative and use that connection to compute the definite integral. The result that I am eventually going to prove sits at the end of a chain of earlier definitions and intermediate results. 2 Some important facts about continuous functions The first intermediate result we are going to have to prove along the way depends on some definitions and theorems concerning continuous functions. Here are those definitions and theorems. The definition of continuity A function f(x) is continuous at a point x = a if the following hold 1. f(a) exists 2. lim f(x) exists xœa 3. lim f(x) = f(a) xœa 1 A function f(x) is continuous in an interval [a,b] if it is continuous at every point in that interval. The extreme value theorem Let f(x) be a continuous function in an interval [a,b].
    [Show full text]
  • Cardinality of Sets
    Cardinality of Sets MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Cardinality of Sets Fall 2014 1 / 15 Outline 1 Sets with Equal Cardinality 2 Countable and Uncountable Sets MAT231 (Transition to Higher Math) Cardinality of Sets Fall 2014 2 / 15 Sets with Equal Cardinality Definition Two sets A and B have the same cardinality, written jAj = jBj, if there exists a bijective function f : A ! B. If no such bijective function exists, then the sets have unequal cardinalities, that is, jAj 6= jBj. Another way to say this is that jAj = jBj if there is a one-to-one correspondence between the elements of A and the elements of B. For example, to show that the set A = f1; 2; 3; 4g and the set B = {♠; ~; }; |g have the same cardinality it is sufficient to construct a bijective function between them. 1 2 3 4 ♠ ~ } | MAT231 (Transition to Higher Math) Cardinality of Sets Fall 2014 3 / 15 Sets with Equal Cardinality Consider the following: This definition does not involve the number of elements in the sets. It works equally well for finite and infinite sets. Any bijection between the sets is sufficient. MAT231 (Transition to Higher Math) Cardinality of Sets Fall 2014 4 / 15 The set Z contains all the numbers in N as well as numbers not in N. So maybe Z is larger than N... On the other hand, both sets are infinite, so maybe Z is the same size as N... This is just the sort of ambiguity we want to avoid, so we appeal to the definition of \same cardinality." The answer to our question boils down to \Can we find a bijection between N and Z?" Does jNj = jZj? True or false: Z is larger than N.
    [Show full text]
  • 5.7 Inverses and Radical Functions Finding the Inverse Of
    SECTION 5.7 iNverses ANd rAdicAl fuNctioNs 435 leARnIng ObjeCTIveS In this section, you will: • Find the inverse of an invertible polynomial function. • Restrict the domain to find the inverse of a polynomial function. 5.7 InveRSeS And RAdICAl FUnCTIOnS A mound of gravel is in the shape of a cone with the height equal to twice the radius. Figure 1 The volume is found using a formula from elementary geometry. __1 V = πr 2 h 3 __1 = πr 2(2r) 3 __2 = πr 3 3 We have written the volume V in terms of the radius r. However, in some cases, we may start out with the volume and want to find the radius. For example: A customer purchases 100 cubic feet of gravel to construct a cone shape mound with a height twice the radius. What are the radius and height of the new cone? To answer this question, we use the formula ____ 3 3V r = ___ √ 2π This function is the inverse of the formula for V in terms of r. In this section, we will explore the inverses of polynomial and rational functions and in particular the radical functions we encounter in the process. Finding the Inverse of a Polynomial Function Two functions f and g are inverse functions if for every coordinate pair in f, (a, b), there exists a corresponding coordinate pair in the inverse function, g, (b, a). In other words, the coordinate pairs of the inverse functions have the input and output interchanged. Only one-to-one functions have inverses. Recall that a one-to-one function has a unique output value for each input value and passes the horizontal line test.
    [Show full text]