Diagonalization and the Pigeonhole Principle
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Chapter 1. Combinatorial Theory 1.3: No More Counting Please 1.3.1
Chapter 1. Combinatorial Theory 1.3: No More Counting Please Slides (Google Drive) Alex Tsun Video (YouTube) In this section, we don't really have a nice successive ordering where one topic leads to the next as we did earlier. This section serves as a place to put all the final miscellaneous but useful concepts in counting. 1.3.1 Binomial Theorem n We talked last time about binomial coefficients of the form k . Today, we'll see how they are used to prove the binomial theorem, which we'll use more later on. For now, we'll see how they can be used to expand (possibly large) exponents below. You may have learned this technique of FOIL (first, outer, inner, last) for expanding (x + y)2. We then combine like-terms (xy and yx). (x + y)2 = (x + y)(x + y) = xx + xy + yx + yy [FOIL] = x2 + 2xy + y2 But, let's say that we wanted to do this for a binomial raised to some higher power, say (x + y)4. There would be a lot more terms, but we could use a similar approach. 1 2 Probability & Statistics with Applications to Computing 1.3 (x + y)4 = (x + y)(x + y)(x + y)(x + y) = xxxx + yyyy + xyxy + yxyy + ::: But what are the terms exactly that are included in this expression? And how could we combine the like-terms though? Notice that each term will be a mixture of x's and y's. In fact, each term will be in the form xkyn−k (in this case n = 4). -
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. -
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. -
Lecture 6: Entropy
Matthew Schwartz Statistical Mechanics, Spring 2019 Lecture 6: Entropy 1 Introduction In this lecture, we discuss many ways to think about entropy. The most important and most famous property of entropy is that it never decreases Stot > 0 (1) Here, Stot means the change in entropy of a system plus the change in entropy of the surroundings. This is the second law of thermodynamics that we met in the previous lecture. There's a great quote from Sir Arthur Eddington from 1927 summarizing the importance of the second law: If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equationsthen so much the worse for Maxwell's equations. If it is found to be contradicted by observationwell these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of ther- modynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation. Another possibly relevant quote, from the introduction to the statistical mechanics book by David Goodstein: Ludwig Boltzmann who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. There are many ways to dene entropy. All of them are equivalent, although it can be hard to see. In this lecture we will compare and contrast dierent denitions, building up intuition for how to think about entropy in dierent contexts. The original denition of entropy, due to Clausius, was thermodynamic. -
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. -
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. -
The Pigeonhole Principle
The Pigeonhole Principle The pigeonhole principle is the following: If m objects are placed into n bins, where m > n, then some bin contains at least two objects. (We proved this in Lecture #02) Why This Matters ● The pigeonhole principle can be used to show a surprising number of results must be true because they are “too big to fail.” ● Given a large enough number of objects with a bounded number of properties, eventually at least two of them will share a property. ● The applications are extremely deep and thought-provoking. Using the Pigeonhole Principle ● To use the pigeonhole principle: ● Find the m objects to distribute. ● Find the n < m buckets into which to distribute them. ● Conclude by the pigeonhole principle that there must be two objects in some bucket. ● The details of how to proceeds from there are specific to the particular proof you're doing. Theorem: For any natural number n, there is a nonzero multiple of n whose digits are all 0s and 1s. Theorem: For any natural number n, there is a nonzero multiple of n whose digits are all 0s and 1s. 1 11 111 1111 11111 There are 10 objects here. 111111 1111111 11111111 111111111 1111111111 Theorem: For any natural number n, there is a nonzero multiple of n whose digits are all 0s and 1s. 0 1 11 1 111 2 1111 11111 3 111111 4 1111111 11111111 5 111111111 6 1111111111 7 8 Theorem: For any natural number n, there is a nonzero multiple of n whose digits are all 0s and 1s. -
Formalization of Some Central Theorems in Combinatorics of Finite
Kalpa Publications in Computing Volume 1, 2017, Pages 43–57 LPAR-21S: IWIL Workshop and LPAR Short Presentations Formalization of some central theorems in combinatorics of finite sets Abhishek Kr Singh School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai [email protected] Abstract We present fully formalized proofs of some central theorems from combinatorics. These are Dilworth’s decomposition theorem, Mirsky’s theorem, Hall’s marriage theorem and the Erdős-Szekeres theorem. Dilworth’s decomposition theorem is the key result among these. It states that in any finite partially ordered set (poset), the size of a smallest chain cover and a largest antichain are the same. Mirsky’s theorem is a dual of Dilworth’s decomposition theorem, which states that in any finite poset, the size of a smallest antichain cover and a largest chain are the same. We use Dilworth’s theorem in the proofs of Hall’s Marriage theorem and the Erdős-Szekeres theorem. The combinatorial objects involved in these theorems are sets and sequences. All the proofs are formalized in the Coq proof assistant. We develop a library of definitions and facts that can be used as a framework for formalizing other theorems on finite posets. 1 Introduction Formalization of any mathematical theory is a difficult task because the length of a formal proof blows up significantly. In combinatorics the task becomes even more difficult due to the lack of structure in the theory. Some statements often admit more than one proof using completely different ideas. Thus, exploring dependencies among important results may help in identifying an effective order amongst them. -
On Cantor's Uncountability Proofs
On Cantor's important proofs W. Mueckenheim University of Applied Sciences, Baumgartnerstrasse 16, D-86161 Augsburg, Germany [email protected] ________________________________________________________ Abstract. It is shown that the pillars of transfinite set theory, namely the uncountability proofs, do not hold. (1) Cantor's first proof of the uncountability of the set of all real numbers does not apply to the set of irrational numbers alone, and, therefore, as it stands, supplies no distinction between the uncountable set of irrational numbers and the countable set of rational numbers. (2) As Cantor's second uncountability proof, his famous second diagonalization method, is an impossibility proof, a simple counter-example suffices to prove its failure. (3) The contradiction of any bijection between a set and its power set is a consequence of the impredicative definition involved. (4) In an appendix it is shown, by a less important proof of Cantor, how transfinite set theory can veil simple structures. 1. Introduction Two finite sets have the same cardinality if there exists a one-to-one correspondence or bijection between them. Cantor extrapolated this theorem to include infinite sets as well: If between a set M and the set Ù of all natural numbers a bijection M ¨ Ù can be established, then M is denumerable or 1 countable and it has the same cardinality as Ù, namely ¡0. There may be many non-bijective mappings, but at least one bijective mapping must exist. In 1874 Cantor [1] published the proof that the set ¿ of all algebraic numbers (including the set – of all rational numbers) is denumerable.