Transition to Mathematical Proofs Chapter 3 - Functions Assignment Solutions
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Lesson 1.2 – Linear Functions Y M a Linear Function Is a Rule for Which Each Unit 1 Change in Input Produces a Constant Change in Output
Lesson 1.2 – Linear Functions y m A linear function is a rule for which each unit 1 change in input produces a constant change in output. m 1 The constant change is called the slope and is usually m 1 denoted by m. x 0 1 2 3 4 Slope formula: If (x1, y1)and (x2 , y2 ) are any two distinct points on a line, then the slope is rise y y y m 2 1 . (An average rate of change!) run x x x 2 1 Equations for lines: Slope-intercept form: y mx b m is the slope of the line; b is the y-intercept (i.e., initial value, y(0)). Point-slope form: y y0 m(x x0 ) m is the slope of the line; (x0, y0 ) is any point on the line. Domain: All real numbers. Graph: A line with no breaks, jumps, or holes. (A graph with no breaks, jumps, or holes is said to be continuous. We will formally define continuity later in the course.) A constant function is a linear function with slope m = 0. The graph of a constant function is a horizontal line, and its equation has the form y = b. A vertical line has equation x = a, but this is not a function since it fails the vertical line test. Notes: 1. A positive slope means the line is increasing, and a negative slope means it is decreasing. 2. If we walk from left to right along a line passing through distinct points P and Q, then we will experience a constant steepness equal to the slope of the line. -
17 Axiom of Choice
Math 361 Axiom of Choice 17 Axiom of Choice De¯nition 17.1. Let be a nonempty set of nonempty sets. Then a choice function for is a function f sucFh that f(S) S for all S . F 2 2 F Example 17.2. Let = (N)r . Then we can de¯ne a choice function f by F P f;g f(S) = the least element of S: Example 17.3. Let = (Z)r . Then we can de¯ne a choice function f by F P f;g f(S) = ²n where n = min z z S and, if n = 0, ² = min z= z z = n; z S . fj j j 2 g 6 f j j j j j 2 g Example 17.4. Let = (Q)r . Then we can de¯ne a choice function f as follows. F P f;g Let g : Q N be an injection. Then ! f(S) = q where g(q) = min g(r) r S . f j 2 g Example 17.5. Let = (R)r . Then it is impossible to explicitly de¯ne a choice function for . F P f;g F Axiom 17.6 (Axiom of Choice (AC)). For every set of nonempty sets, there exists a function f such that f(S) S for all S . F 2 2 F We say that f is a choice function for . F Theorem 17.7 (AC). If A; B are non-empty sets, then the following are equivalent: (a) A B ¹ (b) There exists a surjection g : B A. ! Proof. (a) (b) Suppose that A B. -
9 Power and Polynomial Functions
Arkansas Tech University MATH 2243: Business Calculus Dr. Marcel B. Finan 9 Power and Polynomial Functions A function f(x) is a power function of x if there is a constant k such that f(x) = kxn If n > 0, then we say that f(x) is proportional to the nth power of x: If n < 0 then f(x) is said to be inversely proportional to the nth power of x. We call k the constant of proportionality. Example 9.1 (a) The strength, S, of a beam is proportional to the square of its thickness, h: Write a formula for S in terms of h: (b) The gravitational force, F; between two bodies is inversely proportional to the square of the distance d between them. Write a formula for F in terms of d: Solution. 2 k (a) S = kh ; where k > 0: (b) F = d2 ; k > 0: A power function f(x) = kxn , with n a positive integer, is called a mono- mial function. A polynomial function is a sum of several monomial func- tions. Typically, a polynomial function is a function of the form n n−1 f(x) = anx + an−1x + ··· + a1x + a0; an 6= 0 where an; an−1; ··· ; a1; a0 are all real numbers, called the coefficients of f(x): The number n is a non-negative integer. It is called the degree of the polynomial. A polynomial of degree zero is just a constant function. A polynomial of degree one is a linear function, of degree two a quadratic function, etc. The number an is called the leading coefficient and a0 is called the constant term. -
Equivalents to the Axiom of Choice and Their Uses A
EQUIVALENTS TO THE AXIOM OF CHOICE AND THEIR USES A Thesis Presented to The Faculty of the Department of Mathematics California State University, Los Angeles In Partial Fulfillment of the Requirements for the Degree Master of Science in Mathematics By James Szufu Yang c 2015 James Szufu Yang ALL RIGHTS RESERVED ii The thesis of James Szufu Yang is approved. Mike Krebs, Ph.D. Kristin Webster, Ph.D. Michael Hoffman, Ph.D., Committee Chair Grant Fraser, Ph.D., Department Chair California State University, Los Angeles June 2015 iii ABSTRACT Equivalents to the Axiom of Choice and Their Uses By James Szufu Yang In set theory, the Axiom of Choice (AC) was formulated in 1904 by Ernst Zermelo. It is an addition to the older Zermelo-Fraenkel (ZF) set theory. We call it Zermelo-Fraenkel set theory with the Axiom of Choice and abbreviate it as ZFC. This paper starts with an introduction to the foundations of ZFC set the- ory, which includes the Zermelo-Fraenkel axioms, partially ordered sets (posets), the Cartesian product, the Axiom of Choice, and their related proofs. It then intro- duces several equivalent forms of the Axiom of Choice and proves that they are all equivalent. In the end, equivalents to the Axiom of Choice are used to prove a few fundamental theorems in set theory, linear analysis, and abstract algebra. This paper is concluded by a brief review of the work in it, followed by a few points of interest for further study in mathematics and/or set theory. iv ACKNOWLEDGMENTS Between the two department requirements to complete a master's degree in mathematics − the comprehensive exams and a thesis, I really wanted to experience doing a research and writing a serious academic paper. -
Polynomials.Pdf
POLYNOMIALS James T. Smith San Francisco State University For any real coefficients a0,a1,...,an with n $ 0 and an =' 0, the function p defined by setting n p(x) = a0 + a1 x + AAA + an x for all real x is called a real polynomial of degree n. Often, we write n = degree p and call an its leading coefficient. The constant function with value zero is regarded as a real polynomial of degree –1. For each n the set of all polynomials with degree # n is closed under addition, subtraction, and multiplication by scalars. The algebra of polynomials of degree # 0 —the constant functions—is the same as that of real num- bers, and you need not distinguish between those concepts. Polynomials of degrees 1 and 2 are called linear and quadratic. Polynomial multiplication Suppose f and g are nonzero polynomials of degrees m and n: m f (x) = a0 + a1 x + AAA + am x & am =' 0, n g(x) = b0 + b1 x + AAA + bn x & bn =' 0. Their product fg is a nonzero polynomial of degree m + n: m+n f (x) g(x) = a 0 b 0 + AAA + a m bn x & a m bn =' 0. From this you can deduce the cancellation law: if f, g, and h are polynomials, f =' 0, and fg = f h, then g = h. For then you have 0 = fg – f h = f ( g – h), hence g – h = 0. Note the analogy between this wording of the cancellation law and the corresponding fact about multiplication of numbers. One of the most useful results about integer multiplication is the unique factoriza- tion theorem: for every integer f > 1 there exist primes p1 ,..., pm and integers e1 em e1 ,...,em > 0 such that f = pp1 " m ; the set consisting of all pairs <pk , ek > is unique. -
Bijections and Infinities 1 Key Ideas
INTRODUCTION TO MATHEMATICAL REASONING Worksheet 6 Bijections and Infinities 1 Key Ideas This week we are going to focus on the notion of bijection. This is a way we have to compare two different sets, and say that they have \the same number of elements". Of course, when sets are finite everything goes smoothly, but when they are not... things get spiced up! But let us start, as usual, with some definitions. Definition 1. A bijection between a set X and a set Y consists of matching elements of X with elements of Y in such a way that every element of X has a unique match, and every element of Y has a unique match. Example 1. Let X = f1; 2; 3g and Y = fa; b; cg. An example of a bijection between X and Y consists of matching 1 with a, 2 with b and 3 with c. A NOT example would match 1 and 2 with a and 3 with c. In this case, two things fail: a has two partners, and b has no partner. Note: if you are familiar with these concepts, you may have recognized a bijection as a function between X and Y which is both injective (1:1) and surjective (onto). We will revisit the notion of bijection in this language in a few weeks. For now, we content ourselves of a slightly lower level approach. However, you may be a little unhappy with the use of the word \matching" in a mathematical definition. Here is a more formal definition for the same concept. -
Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Boise State University, [email protected]
Boise State University ScholarWorks Mathematics Undergraduate Theses Department of Mathematics 5-2014 Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Boise State University, [email protected] Follow this and additional works at: http://scholarworks.boisestate.edu/ math_undergraduate_theses Part of the Set Theory Commons Recommended Citation Rahim, Farighon Abdul, "Axioms of Set Theory and Equivalents of Axiom of Choice" (2014). Mathematics Undergraduate Theses. Paper 1. Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Advisor: Samuel Coskey Boise State University May 2014 1 Introduction Sets are all around us. A bag of potato chips, for instance, is a set containing certain number of individual chip’s that are its elements. University is another example of a set with students as its elements. By elements, we mean members. But sets should not be confused as to what they really are. A daughter of a blacksmith is an element of a set that contains her mother, father, and her siblings. Then this set is an element of a set that contains all the other families that live in the nearby town. So a set itself can be an element of a bigger set. In mathematics, axiom is defined to be a rule or a statement that is accepted to be true regardless of having to prove it. In a sense, axioms are self evident. In set theory, we deal with sets. Each time we state an axiom, we will do so by considering sets. Example of the set containing the blacksmith family might make it seem as if sets are finite. -
DISCONTINUITY SETS of SOME INTERVAL BIJECTIONS E. De
International Journal of Pure and Applied Mathematics Volume 79 No. 1 2012, 47-55 AP ISSN: 1311-8080 (printed version) url: http://www.ijpam.eu ijpam.eu DISCONTINUITY SETS OF SOME INTERVAL BIJECTIONS E. de Amo1 §, M. D´ıaz Carrillo2, J. Fern´andez S´anchez3 1,3Departamento de Algebra´ y An´alisis Matem´atico Universidad de Almer´ıa 04120, Almer´ıa, SPAIN 2Departamento de An´alisis Matem´atico Universidad de Granada 18071, Granada, SPAIN Abstract: In this paper we prove that the discontinuity set of any bijective function from [a, b[ to [c, d] is infinite. We then proceed to give a gallery of examples which shows that this is the best we can expect, and that it is possible that any intermediate case exists. We conclude with an example of a (nowhere continuous) bijective function which exhibits a dense graph in the rectangle [a, b[ × [c, d] . AMS Subject Classification: 26A06, 26A09 Key Words: discontinuity set, fixed point 1. Introduction During our investigations [1] concerning on the application of Ergodic Theory of Numbers to generate singular functions (i.e., their derivatives vanish almost everywhere), we came across a bijection from a closed interval to another that it is not closed: c : [0, 1] −→ [0, 1[, given by (segments on some intervals): (n − 1)! n! − 1 1 c (x) := x − + n n2 n + 1 1 1 1 1 which maps 1 − n! , 1 − (n+1)! onto n+1 , n and c(1) = 0. h h h h c 2012 Academic Publications, Ltd. Received: April 19, 2012 url: www.acadpubl.eu §Correspondence author 48 E. -
Real Zeros of Polynomial Functions
333353_0200.qxp 1/11/07 2:00 PM Page 91 Polynomial and Chapter 2 Rational Functions y y y 2.1 Quadratic Functions 2 2 2 2.2 Polynomial Functions of Higher Degree x x x −44−2 −44−2 −44−2 2.3 Real Zeros of Polynomial Functions 2.4 Complex Numbers 2.5 The Fundamental Theorem of Algebra Polynomial and rational functions are two of the most common types of functions 2.6 Rational Functions and used in algebra and calculus. In Chapter 2, you will learn how to graph these types Asymptotes of functions and how to find the zeros of these functions. 2.7 Graphs of Rational Functions 2.8 Quadratic Models David Madison/Getty Images Selected Applications Polynomial and rational functions have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. ■ Automobile Aerodynamics, Exercise 58, page 101 ■ Revenue, Exercise 93, page 114 ■ U.S. Population, Exercise 91, page 129 ■ Impedance, Exercises 79 and 80, page 138 ■ Profit, Exercise 64, page 145 ■ Data Analysis, Exercises 41 and 42, page 154 ■ Wildlife, Exercise 43, page 155 ■ Comparing Models, Exercise 85, page 164 ■ Media, Aerodynamics is crucial in creating racecars.Two types of racecars designed and built Exercise 18, page 170 by NASCAR teams are short track cars, as shown in the photo, and super-speedway (long track) cars. Both types of racecars are designed either to allow for as much downforce as possible or to reduce the amount of drag on the racecar. 91 333353_0201.qxp 1/11/07 2:02 PM Page 92 92 Chapter 2 Polynomial and Rational Functions 2.1 Quadratic Functions The Graph of a Quadratic Function What you should learn Analyze graphs of quadratic functions. -
Cardinality and the Nature of Infinity
Cardinality and The Nature of Infinity Recap from Last Time Functions ● A function f is a mapping such that every value in A is associated with a single value in B. ● For every a ∈ A, there exists some b ∈ B with f(a) = b. ● If f(a) = b0 and f(a) = b1, then b0 = b1. ● If f is a function from A to B, we call A the domain of f andl B the codomain of f. ● We denote that f is a function from A to B by writing f : A → B Injective Functions ● A function f : A → B is called injective (or one-to-one) iff each element of the codomain has at most one element of the domain associated with it. ● A function with this property is called an injection. ● Formally: If f(x0) = f(x1), then x0 = x1 ● An intuition: injective functions label the objects from A using names from B. Surjective Functions ● A function f : A → B is called surjective (or onto) iff each element of the codomain has at least one element of the domain associated with it. ● A function with this property is called a surjection. ● Formally: For any b ∈ B, there exists at least one a ∈ A such that f(a) = b. ● An intuition: surjective functions cover every element of B with at least one element of A. Bijections ● A function that associates each element of the codomain with a unique element of the domain is called bijective. ● Such a function is a bijection. ● Formally, a bijection is a function that is both injective and surjective. -
Cantor's Diagonal Argument
Cantor's diagonal argument such that b3 =6 a3 and so on. Now consider the infinite decimal expansion b = 0:b1b2b3 : : :. Clearly 0 < b < 1, and b does not end in an infinite string of 9's. So b must occur somewhere All of the infinite sets we have seen so far have been `the same size'; that is, we have in our list above (as it represents an element of I). Therefore there exists n 2 N such that N been able to find a bijection from into each set. It is natural to ask if all infinite sets have n 7−! b, and hence we must have the same cardinality. Cantor showed that this was not the case in a very famous argument, known as Cantor's diagonal argument. 0:an1an2an3 : : : = 0:b1b2b3 : : : We have not yet seen a formal definition of the real numbers | indeed such a definition But a =6 b (by construction) and so we cannot have the above equality. This gives the is rather complicated | but we do have an intuitive notion of the reals that will do for our nn n desired contradiction, and thus proves the theorem. purposes here. We will represent each real number by an infinite decimal expansion; for example 1 = 1:00000 : : : Remarks 1 2 = 0:50000 : : : 1 3 = 0:33333 : : : (i) The above proof is known as the diagonal argument because we constructed our π b 4 = 0:78539 : : : element by considering the diagonal elements in the array (1). The only way that two distinct infinite decimal expansions can be equal is if one ends in an (ii) It is possible to construct ever larger infinite sets, and thus define a whole hierarchy infinite string of 0's, and the other ends in an infinite sequence of 9's. -
Chapter 1 Linear Functions
Chapter 1 Linear Functions 11 Sec. 1.1: Slopes and Equations of Lines Date Lines play a very important role in Calculus where we will be approximating complicated functions with lines. We need to be experts with lines to do well in Calculus. In this section, we review slope and equations of lines. Slope of a Line: The slope of a line is defined as the vertical change (the \rise") over the horizontal change (the \run") as one travels along the line. In symbols, taking two different points (x1; y1) and (x2; y2) on the line, the slope is Change in y ∆y y − y m = = = 2 1 : Change in x ∆x x2 − x1 Example: World milk production rose at an approximately constant rate between 1996 and 2003 as shown in the following graph: where M is in million tons and t is the years since 1996. Estimate the slope and interpret it in terms of milk production. 12 Clicker Question 1: Find the slope of the line through the following pair of points (−2; 11) and (3; −4): The slope is (A) Less than -2 (B) Between -2 and 0 (C) Between 0 and 2 (D) More than 2 (E) Undefined It will be helpful to recall the following facts about intercepts. Intercepts: x-intercepts The points where a graph touches the x-axis. If we have an equation, we can find them by setting y = 0. y-intercepts The points where a graph touches the y-axis. If we have an equation, we can find them by setting x = 0.