Pascal s Triangle and III.3 Answers
|
|
- Melanie Nash
- 5 years ago
- Views:
Transcription
1 Chapter 16 Pascal s Triangle and III3 Answers Pascal s Triangle Consider the pattern of numbers: This pattern is called Pascal s Triangle, and it goes on forever The best mathematical way to describe it is: C 0,c δ 0,c for all c Z C r,c C r 1,c 1 + C r 1,c for r>0,c Z The elements C 0,c go in the first row of the table Only the element C 0,0 1is shown, the rest (all zeros) are suppressed The element C r,c is element c in row r (counting started from 0 not 1, soc 3,1 3) For the most part the following problems are good exercises in mathematical induction Each of the following proofs uses the previous results 1 If r 0 and c<0 or c>rthen C r,c 0 Proof If r 0then by definition C 0,c 0if c 6 0 That is, C 0,c 0if c<0or ċ>0 For the induction step, suppose C r 1,c 0if c<0or c>r 1 We want to show that C r,c 0if c<0 or c>r We know Ċr,c C r 1,c 1 + C r 1,c If c<0 then, since r 1 <rand c 1 < 0, C r 1,c 1 C r 1,c 0so C r,c 0Ifċ>rthen, since r 1 >c 1 we have C r 1,c 1 C r 1,c 0so C r,c 0 2 C r,0 C r,r 1all r 0 Proof Again, if r 0, C r,0 C r,r C 0,0 1 For the induction step, suppose C r 1,0 C r 1,r 1 1 Then C r,0 C r 1, 1 + C r 1, C r,r C r 1,r 1 + C r 1,r
2 3 C r,1 C r,r 1 r, allr 0 (Check the cases r 0and r 1first, then use induction for the rest) Proof Examining the table above, we see for r 1that C r,1 C 0,1 0r C r,r 1 C 0, 1 0r For r 1 C r,1 C 1,1 1r C r,r 1 C 1,0 1r Now for the induction step: C r,1 C r 1,0 + C r 1,1 1+(r 1) r C r,r 1 C r 1,r 1 + C r 1,r 1 (r 1) + 1 r 4 Develop a formula for the sum of the elements in row r and prove it Theorem 38 P r i0 C r,i 2 r Proof If r 0then rx C r,i Ċ0, i0 Now for the induction step Note that since C r,c 0for c<0 and ċ>r,wemayrestatethetheorem as C r,i 2 r To prove this for r>0 we may assume C r,i C r 1,i 2 r 1 : (C r 1,i 1 + C r 1,i ) C r 1,i 1 + C r 1,i + 2 r 1 +2 r 1 2 r C r 1,i C r 1,i 5 Find out what C r,0 C r,1 + C r,2 ±C r,r equals, and prove it Theorem 39 P ½ 1 if r 0 ( 1)i C r,i 0 if r>0 31
3 Proof If r 0then P ( 1)i C r,i C 0,0 1Forr>0 we use induction ( 1) i C r,i 0 ( 1) i (C r 1,i 1 + C r 1,i ) ( 1) i C r 1,i 1 + ( 1) i 1 C r 1,i 1 + ( 1) i C r 1,i + ( 1) i C r 1,i ( 1) i C r 1,i ( 1) i C r 1,i The numbers C r,c show up in two important mathematical contexts Usually we denote C r,c by µ r C r,c c the binomial coefficient It represents the number of subsets of size c in a set of size r 6 Verify that the number of subsets of size 5 in a set of size 8 is C 8,5 Proof C 8,5 56, and the 56 subsets of size 5 from {1, 2, 3, 4, 5, 6, 7, 8}are {1, 2, 3, 4, 5}, {1, 2, 3, 4, 6}, {1, 2, 3, 4, 7}, {1, 2, 3, 4, 8}, {1, 2, 3, 5, 6}, {1, 2, 3, 5, 7}, {1, 2, 3, 5, 8}, {1, 2, 3, 6, 7} and so forth These numbers are called binomial coefficients because: (x + y) r C r,0 x r + C r,1 x r 1 y + C r,2 x r 2 y C r,r 2 x 2 y r 2 + C r,r 1 xy r 1 + C r,r y r 7 Verify this formula for r 4Expand(x + y) 4 and show that the coefficients are C 4,0,C 4,1,,C 4,4 Proof C 4,0 C 4,4 1, C 4,1 C 4,3 4 C 4,2 6Moreover(x + y) 4 x 4 +4x 3 y+6x 2 y 2 +4xy 3 +y 4 Eccles III3 Suppose 100 people speak one or more of the languages English, Spanish and Swahili The number speaking English (and possibly Spanish or Swahili) is 75; the number speaking Spanish is 60 and the number speaking Swahili is 45 We wish to answer several questions about the possible linguistic composition of this motley crowd, but first we need to set up some equationsdefine some variables: e # English only s # Spanish only w # Swahili only es # English and Spanish ew # English and Swahili sw # Spanish and Swahili esw # English and Spanish and Swahili 32
4 The given conditions imply: Row-reducing this system, we get: es + ew + sw + esw + e + s + w 100 es + ew + esw + e 75 es + sw + esw + s 60 ew + sw + esw + w 45 es 55 e s ew 40 e w sw 25 s w esw e + s + w 20 Note that solutions fitting the problem require that all variables have non-negative integral values 1 Since es, ew and sw must be non-negative, he first three lines imply e + s 55 e + w 40 s + w 25 Adding up these inequalities and dividing by 2 gives: e + s + w 60 That is, the number of people speaking only one language cannot exceed 60 2 Since ew 0, the second line implies e 40, or the maximum number of people speaking only English is 40 If 40 people speak only English, then from the first equation 15 people speak only Spanish and from the second equation no people speak only Swahili 3 From the third line we see that the number of people who speak only one language (e + s + w) is 20 more than the number of people speaking all three (esw) Thus the more people who speak all three, themorewhospeakonlyone 33
5 Chapter 17 Counting Again General nonsense We begin by reorganizing a proof from the last lecture Lemma 40 If h : N n N n is injective, then h is also surjective Proof We proceed by induction First the base case n 1 Ifwehaveamaph : {1} {1} then the only possible value for h (1) is 1, soh is surjective (For this base case we did not have to assume that h was injective because all maps h : {1} {1} are injective) Now for the inductive step Suppose we have an injective function h : N n N n We want to show that h is surjective We may assume that any injective function k : N n 1 N n 1 is surjective We distinguish threepossiblecases 1 h (n) n 2 There exists j N n 1 such that h (j) n 3 For all j N n, h (j) <n We will show, in case (1) and (2), thath is surjective, and we will show that case (3) leads to a contradiction 1 Since h is injective, i<nimplies h (i) <n Thus the restricted map h : N n 1 N n 1 and h, restricted to N n 1, is still injective By the inductive hypothesis (we have to use it somewhere), the restricted map h : N n 1 N n 1 is surjective Therefore h : N n N n is surjective 2 Define a function k : N n N n by k (j) h (n) k (n) h (j) n k (i) h (i) for i 6 j and i 6 n The function k is 1 1 and satisfies condition (1) Therefore k is surjective, and thus h is surjective 3 The restricted map h : N n 1 N n is actually a map h : N n 1 N n 1 Since the restricted map is injective, it is also surjective But this contradicts the injectivity of h, because there exists j<nsuch that h (j) h (n) Therefore case(3) leads to a contradiction Proposition 41 Suppose we have a bijective function f : N n N m Then n m 34
6 Proof The proof is by contradiction Suppose n 6 m Replacing f by f 1 if necessary, we may suppose n>m Construct g : N m N n,g(i) i Thisisdefined because m<n The map g is injective, but g is not surjective because there is no element x N m, g (x) n Thusg f : N n N n is injective but not surjective, which by the Lemma is a contradiction Lemma 42 Suppose f : N m N n is injective Then m n Proof Suppose m>n We will derive a contradiction Since m>nwe have a map g : N n N m, g (i) i The function g is injective, so g f : N m N m is injective By the previous lemma, g f is surjective, so g is surjective Thus g is bijective and n m, which is a contradiction Proposition 43 (The pigeonhole principle) Suppose A and B are finite sets Then A B if and only if there is an injection f : A B Moreover A B if and only if there exists an injection f : A B and every such injection is bijective Proof Since A and B are finite, there exist bijections g : N m A and h : N n B, wherem A and n B Iff is injective, we have an injective map: h 1 f g : N m N n By the lemma, m n Conversely, if m n we have an injective map k : N m N n, k (i) i Thus f h k g 1 : A B is injective Suppose A B Bythefirstpartwehaveaninjectionf : A B Letk h 1 f g : N m N n This map is injective and n m,so by the first lemma k is bijective Therefore f h k g 1 is a composition of bijective maps and is therefore bijective Conversely, if we have an injection f : A B then by an earlier proposition A and B have the same cardinality or A B Proposition 44 If B is a finite set and A B then A is finite Proof To show A is finite, we proceed by induction on B If B 0then B φ so A φ and A is finite If B 1then B {b} for some element b ThuseitherA φ or A B IneithercaseA is finite We proceed by induction on B, assuming B > 1 Let n B Wehaveabijectionf : N n B Define:b f (n) The map f restricts to a bijection N n 1 B {b}, sob {b} is a finite set, and B {b} < B Also {b} 1 Let A 1 A (B {b}) and A 2 A {b} By induction, A 1 and A 2 are finite sets Also A 1 A 2 φby an earlier result that the disjoint union of finite sets if finite, A A 1 A 2 is finite Corollary 45 If B is a finite set and A B then A B Moreoverif A B then A B Proof Since subsets of B are finite, both A and B A are finite Since B A + B A and all cardinalities are non-negative, A B Moreover if A B then B A 0or B A φ That is, A B Counting Functions Definition 46 Let A and B be sets fun(a, B) is the set of functions with domain A and codomain B, that is the set of functions f : A B Definition 47 The graph of f : A B is the subset {(a, f (a)) : a A} A B Proposition 48 If two functions have the same graph, they are the same function Thus fun(a, B) P (A B) Proposition 49 If A and B are non-empty finite sets then the number of functions f : A B is B A 35
7 Proof It suffices to show that the number of functions N m N n is n m The intuition is that function f : N m N n is constructed by choosing a value for f (1), avaluefor f (2) andsoonthroughf (m) Each value can be selected n ways, so overall n m choices are possible But intuition is not proof The proof goes by induction on m Ifm 1we must count the number of functions {1} {1, 2,,n} There are n such functions given by f (1) i for 1 i n Thus if m 1the number of functions is n m n 1 n If m>1, then by induction there are n m 1 functions N m 1 N n For 1 i n, define F i {f fun(n m, N n ):f (m) i} There is a 1 1 correspondence F i fun(n m 1, N n ),so F i n m 1 Moreover, fun(n m, N n ) is the disjoint union of the F i,so fun(n m, N n ) nx F i i1 n n m 1 n m Definition 50 A permutation of a set A is a bijective function f : A A Proposition 51 If A is a finite set with A n then the number of permutations of A is n! Remark 52 0! 1 because there is one permutation of the empty set Proof It suffices to prove that the number of permutations of N n is n!if n 1there is obviously 1 (bijective) map from N 1 N 1 We go by induction on n, since and we can assume that the number of bijectiions between two sets of size n 1 is (n 1)! Let P be the set of permutaions of N n,andlet P i {p P : p (n) i} ThenP is the disjoint union of the P i, 1 i n, andp i is in 1 1 correspondence with the set of bijective maps N n 1 (N n {i}) Both these sets of n 1 elements, so P i (n 1)! Thus P n (n 1)! n! Definition 53 C n,r is the number defined in the assignment "Pascal s Triangle" D n,r is the number of subsets of cardinality r is a set of cardinality n n (n 1) (n r +1) if 0 <r n! r! if 0 r n r 1 if r 0 r!(n r)! 0 otherwise 0 if r<0 P n,r coefficient of x n r y r in (x + y) n Proposition 54 The four functions defined above are all equal Proof Note that all the functions are defined for n 0 and r Z To show all the functions are equal, it suffices to show that all the other functions are equal to C n,r The values of the function C n,r are determined by the relations: C 0,r δ 0,r C n,r C n 1,r 1 + C n 1,r for n>0 and all r Thus, to show that a function f (n, r) C n,r,itsuffices to show f (0,r) δ 0,r f (n, r) f (n 1,r 1) + f (n 1,r) for n>0 and all r We begin with D n,r First of all, D 0,r is the number of subsets of cardinality r in the empty set The empty set has only one subset, namely itself, so D 0,0 1and D 0,r 0for r 6 0 If n>0 let A be a set of cardinality n Choose an element a A (possible because the cardinality of A is not 0), and let B A {a} The subsets of cardinality r in A fall into two categories, those contained in B 36
8 and those not contained in BThe number of subsets of cardinality r contained in B is D n 1,r Thesubsets not contained in B are the subsets containing a The number of subsets of A of cardinality r containing a is the number of subsets of B of cardinality r 1, ord n 1,r 1 Thus the number of subets of A of cardinality r is D n,r D n 1,r 1 + D n 1,r µ 0 Now we consider If n 0then δ r,0 If n>0then we must show µ r r r r n 1 + There are four cases to consider: r µ µ n 1 n 1 1 If r<0 then r r 1 r µ µ µ µ n 1 n 1 n 1 n 1 2 If r 0then r r 1 r 3 If r 1then n r n 1 µ µ µ µ n 1 n 1 n 1 n r 1 r µ n 1 r 1 4 If r>1 then r n (n 1) (n r +1) r! (n 1) (n r +1) n (r 1)! r (n 1) (n r +1) (r 1)! µ 1+ n r r (n 1) ((n 1) (r 1) + 1) + (r 1)! µ µ n 1 n 1 + r 1 r (n 1) ((n 1) r +1) r! Finally we look at P n,r First of all (x + y) 0 1x 0 y 0 so P 0,0 1and P 0,r 0for r 6 0Ifn>0 then (x + y) n nx P n,r x n r y r r0 (x + y)(x + y) n 1 n1 X (x + y) P n 1,r x n 1 r y r r0 r0 n1 X n1 X P n 1,r x n r y r + P n 1,r x n 1 r y r+1 n1 X P n 1,r x n r y r + r0 r0 nx P n 1,r 1 x n r y r r1 nx (P n 1,r 1 + P n 1,r ) x n r y r (because P n 1, 1 P n 1,n 0) r0 Thus P n,r P n 1,r 1 + P n 1,r 37
6 CARDINALITY OF SETS
6 CARDINALITY OF SETS MATH10111 - Foundations of Pure Mathematics We all have an idea of what it means to count a finite collection of objects, but we must be careful to define rigorously what it means
More informationChapter 1 : The language of mathematics.
MAT 200, Logic, Language and Proof, Fall 2015 Summary Chapter 1 : The language of mathematics. Definition. A proposition is a sentence which is either true or false. Truth table for the connective or :
More informationPostulate 2 [Order Axioms] in WRW the usual rules for inequalities
Number Systems N 1,2,3,... the positive integers Z 3, 2, 1,0,1,2,3,... the integers Q p q : p,q Z with q 0 the rational numbers R {numbers expressible by finite or unending decimal expansions} makes sense
More information5 Set Operations, Functions, and Counting
5 Set Operations, Functions, and Counting Let N denote the positive integers, N 0 := N {0} be the non-negative integers and Z = N 0 ( N) the positive and negative integers including 0, Q the rational numbers,
More information1 Counting Collections of Functions and of Subsets.
1 Counting Collections of Functions and of Subsets See p144 All page references are to PJEccles book unless otherwise stated Let X and Y be sets Definition 11 F un (X, Y will be the set of all functions
More informationBasic counting techniques. Periklis A. Papakonstantinou Rutgers Business School
Basic counting techniques Periklis A. Papakonstantinou Rutgers Business School i LECTURE NOTES IN Elementary counting methods Periklis A. Papakonstantinou MSIS, Rutgers Business School ALL RIGHTS RESERVED
More informationLecture 4: Counting, Pigeonhole Principle, Permutations, Combinations Lecturer: Lale Özkahya
BBM 205 Discrete Mathematics Hacettepe University http://web.cs.hacettepe.edu.tr/ bbm205 Lecture 4: Counting, Pigeonhole Principle, Permutations, Combinations Lecturer: Lale Özkahya Resources: Kenneth
More informationADVANCED CALCULUS - MTH433 LECTURE 4 - FINITE AND INFINITE SETS
ADVANCED CALCULUS - MTH433 LECTURE 4 - FINITE AND INFINITE SETS 1. Cardinal number of a set The cardinal number (or simply cardinal) of a set is a generalization of the concept of the number of elements
More informationLecture Notes 1 Basic Concepts of Mathematics MATH 352
Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM 87801 June 3, 2004 Author: Ivan Avramidi; File: absmath.tex; Date: June 11,
More informationCounting. Mukulika Ghosh. Fall Based on slides by Dr. Hyunyoung Lee
Counting Mukulika Ghosh Fall 2018 Based on slides by Dr. Hyunyoung Lee Counting Counting The art of counting is known as enumerative combinatorics. One tries to count the number of elements in a set (or,
More informationAxioms for Set Theory
Axioms for Set Theory The following is a subset of the Zermelo-Fraenkel axioms for set theory. In this setting, all objects are sets which are denoted by letters, e.g. x, y, X, Y. Equality is logical identity:
More informationEquivalence of Propositions
Equivalence of Propositions 1. Truth tables: two same columns 2. Sequence of logical equivalences: from one to the other using equivalence laws 1 Equivalence laws Table 6 & 7 in 1.2, some often used: Associative:
More informationSets, Structures, Numbers
Chapter 1 Sets, Structures, Numbers Abstract In this chapter we shall introduce most of the background needed to develop the foundations of mathematical analysis. We start with sets and algebraic structures.
More informationDefinition: Let S and T be sets. A binary relation on SxT is any subset of SxT. A binary relation on S is any subset of SxS.
4 Functions Before studying functions we will first quickly define a more general idea, namely the notion of a relation. A function turns out to be a special type of relation. Definition: Let S and T be
More informationMATH 220 (all sections) Homework #12 not to be turned in posted Friday, November 24, 2017
MATH 220 (all sections) Homework #12 not to be turned in posted Friday, November 24, 2017 Definition: A set A is finite if there exists a nonnegative integer c such that there exists a bijection from A
More informationDiscrete Mathematics & Mathematical Reasoning Chapter 6: Counting
Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 39 Chapter Summary The Basics
More informationWell Ordered Sets (continued)
Well Ordered Sets (continued) Theorem 8 Given any two well-ordered sets, either they are isomorphic, or one is isomorphic to an initial segment of the other. Proof Let a,< and b, be well-ordered sets.
More informationLecture 8: Equivalence Relations
Lecture 8: Equivalence Relations 1 Equivalence Relations Next interesting relation we will study is equivalence relation. Definition 1.1 (Equivalence Relation). Let A be a set and let be a relation on
More informationGraph Theory. Thomas Bloom. February 6, 2015
Graph Theory Thomas Bloom February 6, 2015 1 Lecture 1 Introduction A graph (for the purposes of these lectures) is a finite set of vertices, some of which are connected by a single edge. Most importantly,
More informationDiscrete Mathematics: Lectures 6 and 7 Sets, Relations, Functions and Counting Instructor: Arijit Bishnu Date: August 4 and 6, 2009
Discrete Mathematics: Lectures 6 and 7 Sets, Relations, Functions and Counting Instructor: Arijit Bishnu Date: August 4 and 6, 2009 Our main goal is here is to do counting using functions. For that, we
More information4) Have you met any functions during our previous lectures in this course?
Definition: Let X and Y be sets. A function f from the set X to the set Y is a rule which associates to each element x X a unique element y Y. Notation: f : X Y f defined on X with values in Y. x y y =
More informationHarvard CS 121 and CSCI E-207 Lecture 6: Regular Languages and Countability
Harvard CS 121 and CSCI E-207 Lecture 6: Regular Languages and Countability Salil Vadhan September 20, 2012 Reading: Sipser, 1.3 and The Diagonalization Method, pages 174 178 (from just before Definition
More informationMATH 2200 Final Review
MATH 00 Final Review Thomas Goller December 7, 01 1 Exam Format The final exam will consist of 8-10 proofs It will take place on Tuesday, December 11, from 10:30 AM - 1:30 PM, in the usual room Topics
More informationFoundations of Mathematics MATH 220 FALL 2017 Lecture Notes
Foundations of Mathematics MATH 220 FALL 2017 Lecture Notes These notes form a brief summary of what has been covered during the lectures. All the definitions must be memorized and understood. Statements
More informationPart V. Chapter 19. Congruence of integers
Part V. Chapter 19. Congruence of integers Congruence modulo m Let m be a positive integer. Definition. Integers a and b are congruent modulo m if and only if a b is divisible by m. For example, 1. 277
More information0 Sets and Induction. Sets
0 Sets and Induction Sets A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a A to denote that a is an element of the set
More informationBinomial Coefficient Identities/Complements
Binomial Coefficient Identities/Complements CSE21 Fall 2017, Day 4 Oct 6, 2017 https://sites.google.com/a/eng.ucsd.edu/cse21-fall-2017-miles-jones/ permutation P(n,r) = n(n-1) (n-2) (n-r+1) = Terminology
More informationTopological properties
CHAPTER 4 Topological properties 1. Connectedness Definitions and examples Basic properties Connected components Connected versus path connected, again 2. Compactness Definition and first examples Topological
More informationREVIEW FOR THIRD 3200 MIDTERM
REVIEW FOR THIRD 3200 MIDTERM PETE L. CLARK 1) Show that for all integers n 2 we have 1 3 +... + (n 1) 3 < 1 n < 1 3 +... + n 3. Solution: We go by induction on n. Base Case (n = 2): We have (2 1) 3 =
More informationPart IA Numbers and Sets
Part IA Numbers and Sets Theorems Based on lectures by A. G. Thomason Notes taken by Dexter Chua Michaelmas 2014 These notes are not endorsed by the lecturers, and I have modified them (often significantly)
More informationSolution. 1 Solutions of Homework 1. 2 Homework 2. Sangchul Lee. February 19, Problem 1.2
Solution Sangchul Lee February 19, 2018 1 Solutions of Homework 1 Problem 1.2 Let A and B be nonempty subsets of R + :: {x R : x > 0} which are bounded above. Let us define C = {xy : x A and y B} Show
More informationCSCE 222 Discrete Structures for Computing. Review for the Final. Hyunyoung Lee
CSCE 222 Discrete Structures for Computing Review for the Final! Hyunyoung Lee! 1 Final Exam Section 501 (regular class time 8:00am) Friday, May 8, starting at 1:00pm in our classroom!! Section 502 (regular
More informationSolutions to Homework Set 1
Solutions to Homework Set 1 1. Prove that not-q not-p implies P Q. In class we proved that A B implies not-b not-a Replacing the statement A by the statement not-q and the statement B by the statement
More informationBasics of Model Theory
Chapter udf Basics of Model Theory bas.1 Reducts and Expansions mod:bas:red: defn:reduct mod:bas:red: prop:reduct Often it is useful or necessary to compare languages which have symbols in common, as well
More informationMAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9
MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended
More informationIVA S STUDY GUIDE FOR THE DISCRETE FINAL EXAM - SELECTED SOLUTIONS. 1. Combinatorics
IVA S STUDY GUIDE FOR THE DISCRETE FINAL EXAM - SELECTED SOLUTIONS Combinatorics Go over combinatorics examples in the text Review all the combinatorics problems from homewor Do at least a couple of extra
More informationDisjoint Hamiltonian Cycles in Bipartite Graphs
Disjoint Hamiltonian Cycles in Bipartite Graphs Michael Ferrara 1, Ronald Gould 1, Gerard Tansey 1 Thor Whalen Abstract Let G = (X, Y ) be a bipartite graph and define σ (G) = min{d(x) + d(y) : xy / E(G),
More informationThe integers. Chapter 3
Chapter 3 The integers Recall that an abelian group is a set A with a special element 0, and operation + such that x +0=x x + y = y + x x +y + z) =x + y)+z every element x has an inverse x + y =0 We also
More information1.4 Cardinality. Tom Lewis. Fall Term Tom Lewis () 1.4 Cardinality Fall Term / 9
1.4 Cardinality Tom Lewis Fall Term 2006 Tom Lewis () 1.4 Cardinality Fall Term 2006 1 / 9 Outline 1 Functions 2 Cardinality 3 Cantor s theorem Tom Lewis () 1.4 Cardinality Fall Term 2006 2 / 9 Functions
More informationCardinality and ordinal numbers
Cardinality and ordinal numbers The cardinality A of a finite set A is simply the number of elements in it. When it comes to infinite sets, we no longer can speak of the number of elements in such a set.
More informationMATH FINAL EXAM REVIEW HINTS
MATH 109 - FINAL EXAM REVIEW HINTS Answer: Answer: 1. Cardinality (1) Let a < b be two real numbers and define f : (0, 1) (a, b) by f(t) = (1 t)a + tb. (a) Prove that f is a bijection. (b) Prove that any
More informationc i r i i=1 r 1 = [1, 2] r 2 = [0, 1] r 3 = [3, 4].
Lecture Notes: Rank of a Matrix Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk 1 Linear Independence Definition 1. Let r 1, r 2,..., r m
More informationNotes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr.
Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr. Chapter : Logic Topics:. Statements, Negation, and Compound Statements.2 Truth Tables and Logical Equivalences.3
More informationIntroduction to Automata
Introduction to Automata Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjin@postech.ac.kr 1 /
More informationIntroduction to Decision Sciences Lecture 11
Introduction to Decision Sciences Lecture 11 Andrew Nobel October 24, 2017 Basics of Counting Product Rule Product Rule: Suppose that the elements of a collection S can be specified by a sequence of k
More informationTheory of Computation
Thomas Zeugmann Hokkaido University Laboratory for Algorithmics http://www-alg.ist.hokudai.ac.jp/ thomas/toc/ Lecture 1: Introducing Formal Languages Motivation I This course is about the study of a fascinating
More informationDiscussion Summary 10/16/2018
Discussion Summary 10/16/018 1 Quiz 4 1.1 Q1 Let r R and r > 1. Prove the following by induction for every n N, assuming that 0 N as in the book. r 1 + r + r 3 + + r n = rn+1 r r 1 Proof. Let S n = Σ n
More informationPOL502: Foundations. Kosuke Imai Department of Politics, Princeton University. October 10, 2005
POL502: Foundations Kosuke Imai Department of Politics, Princeton University October 10, 2005 Our first task is to develop the foundations that are necessary for the materials covered in this course. 1
More informationChapter 4. Measure Theory. 1. Measure Spaces
Chapter 4. Measure Theory 1. Measure Spaces Let X be a nonempty set. A collection S of subsets of X is said to be an algebra on X if S has the following properties: 1. X S; 2. if A S, then A c S; 3. if
More informationcan only hit 3 points in the codomain. Hence, f is not surjective. For another example, if n = 4
.. Conditions for Injectivity and Surjectivity In this section, we discuss what we can say about linear maps T : R n R m given only m and n. We motivate this problem by looking at maps f : {,..., n} {,...,
More informationTopology. Xiaolong Han. Department of Mathematics, California State University, Northridge, CA 91330, USA address:
Topology Xiaolong Han Department of Mathematics, California State University, Northridge, CA 91330, USA E-mail address: Xiaolong.Han@csun.edu Remark. You are entitled to a reward of 1 point toward a homework
More informationCounting. Math 301. November 24, Dr. Nahid Sultana
Basic Principles Dr. Nahid Sultana November 24, 2012 Basic Principles Basic Principles The Sum Rule The Product Rule Distinguishable Pascal s Triangle Binomial Theorem Basic Principles Combinatorics: The
More informationCOMBINATORIAL COUNTING
COMBINATORIAL COUNTING Our main reference is [1, Section 3] 1 Basic counting: functions and subsets Theorem 11 (Arbitrary mapping Let N be an n-element set (it may also be empty and let M be an m-element
More informationGraphs with few total dominating sets
Graphs with few total dominating sets Marcin Krzywkowski marcin.krzywkowski@gmail.com Stephan Wagner swagner@sun.ac.za Abstract We give a lower bound for the number of total dominating sets of a graph
More information1 Take-home exam and final exam study guide
Math 215 - Introduction to Advanced Mathematics Fall 2013 1 Take-home exam and final exam study guide 1.1 Problems The following are some problems, some of which will appear on the final exam. 1.1.1 Number
More informationCSCE 222 Discrete Structures for Computing. Review for Exam 2. Dr. Hyunyoung Lee !!!
CSCE 222 Discrete Structures for Computing Review for Exam 2 Dr. Hyunyoung Lee 1 Strategy for Exam Preparation - Start studying now (unless have already started) - Study class notes (lecture slides and
More informationCographs; chordal graphs and tree decompositions
Cographs; chordal graphs and tree decompositions Zdeněk Dvořák September 14, 2015 Let us now proceed with some more interesting graph classes closed on induced subgraphs. 1 Cographs The class of cographs
More informationIntroduction to Mathematical Analysis I. Second Edition. Beatriz Lafferriere Gerardo Lafferriere Nguyen Mau Nam
Introduction to Mathematical Analysis I Second Edition Beatriz Lafferriere Gerardo Lafferriere Nguyen Mau Nam Introduction to Mathematical Analysis I Second Edition Beatriz Lafferriere Gerardo Lafferriere
More informationSolutions for Chapter Solutions for Chapter 17. Section 17.1 Exercises
Solutions for Chapter 17 403 17.6 Solutions for Chapter 17 Section 17.1 Exercises 1. Suppose A = {0,1,2,3,4}, B = {2,3,4,5} and f = {(0,3),(1,3),(2,4),(3,2),(4,2)}. State the domain and range of f. Find
More informationMAIN THEOREM OF GALOIS THEORY
MAIN THEOREM OF GALOIS THEORY Theorem 1. [Main Theorem] Let L/K be a finite Galois extension. and (1) The group G = Gal(L/K) is a group of order [L : K]. (2) The maps defined by and f : {subgroups of G}!
More informationMATH 2200 Final LC Review
MATH 2200 Final LC Review Thomas Goller April 25, 2013 1 Final LC Format The final learning celebration will consist of 12-15 claims to be proven or disproven. It will take place on Wednesday, May 1, from
More informationSETS AND FUNCTIONS JOSHUA BALLEW
SETS AND FUNCTIONS JOSHUA BALLEW 1. Sets As a review, we begin by considering a naive look at set theory. For our purposes, we define a set as a collection of objects. Except for certain sets like N, Z,
More informationDiscrete Mathematics 2007: Lecture 5 Infinite sets
Discrete Mathematics 2007: Lecture 5 Infinite sets Debrup Chakraborty 1 Countability The natural numbers originally arose from counting elements in sets. There are two very different possible sizes for
More informationDefinitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations
Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of
More informationFUNCTORS AND ADJUNCTIONS. 1. Functors
FUNCTORS AND ADJUNCTIONS Abstract. Graphs, quivers, natural transformations, adjunctions, Galois connections, Galois theory. 1.1. Graph maps. 1. Functors 1.1.1. Quivers. Quivers generalize directed graphs,
More informationCombinatorics. But there are some standard techniques. That s what we ll be studying.
Combinatorics Problem: How to count without counting. How do you figure out how many things there are with a certain property without actually enumerating all of them. Sometimes this requires a lot of
More informationLecture 10: Everything Else
Math 94 Professor: Padraic Bartlett Lecture 10: Everything Else Week 10 UCSB 2015 This is the tenth week of the Mathematics Subject Test GRE prep course; here, we quickly review a handful of useful concepts
More information2.1 Sets. Definition 1 A set is an unordered collection of objects. Important sets: N, Z, Z +, Q, R.
2. Basic Structures 2.1 Sets Definition 1 A set is an unordered collection of objects. Important sets: N, Z, Z +, Q, R. Definition 2 Objects in a set are called elements or members of the set. A set is
More informationReview 3. Andreas Klappenecker
Review 3 Andreas Klappenecker Final Exam Friday, May 4, 2012, starting at 12:30pm, usual classroom Topics Topic Reading Algorithms and their Complexity Chapter 3 Logic and Proofs Chapter 1 Logic and Proofs
More information1. Foundations of Numerics from Advanced Mathematics. Mathematical Essentials and Notation
1. Foundations of Numerics from Advanced Mathematics Mathematical Essentials and Notation Mathematical Essentials and Notation, October 22, 2012 1 The main purpose of this first chapter (about 4 lectures)
More informationName (please print) Mathematics Final Examination December 14, 2005 I. (4)
Mathematics 513-00 Final Examination December 14, 005 I Use a direct argument to prove the following implication: The product of two odd integers is odd Let m and n be two odd integers Since they are odd,
More informationMidterm Exam. There are 6 problems. Your 5 best answers count. Please pay attention to the presentation of your work! Best 5
Department of Mathematical Sciences Instructor: Daiva Pucinskaite Modern Algebra June 22, 2017 Midterm Exam There are 6 problems. Your 5 best answers count. Please pay attention to the presentation of
More informationarxiv: v1 [math.co] 8 Feb 2014
COMBINATORIAL STUDY OF THE DELLAC CONFIGURATIONS AND THE q-extended NORMALIZED MEDIAN GENOCCHI NUMBERS ANGE BIGENI arxiv:1402.1827v1 [math.co] 8 Feb 2014 Abstract. In two recent papers (Mathematical Research
More information1 The Erdős Ko Rado Theorem
1 The Erdős Ko Rado Theorem A family of subsets of a set is intersecting if any two elements of the family have at least one element in common It is easy to find small intersecting families; the basic
More informationTopics in Logic, Set Theory and Computability
Topics in Logic, Set Theory and Computability Homework Set #3 Due Friday 4/6 at 3pm (by email or in person at 08-3234) Exercises from Handouts 7-C-2 7-E-6 7-E-7(a) 8-A-4 8-A-9(a) 8-B-2 8-C-2(a,b,c) 8-D-4(a)
More informationAN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES
AN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES DUSTIN HEDMARK Abstract. A study of the conditions under which a topological space is metrizable, concluding with a proof of the Nagata Smirnov
More informationTHE REGULAR ELEMENT PROPERTY
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 7, July 1998, Pages 2123 2129 S 0002-9939(98)04257-9 THE REGULAR ELEMENT PROPERTY FRED RICHMAN (Communicated by Wolmer V. Vasconcelos)
More informationPropositional Logic, Predicates, and Equivalence
Chapter 1 Propositional Logic, Predicates, and Equivalence A statement or a proposition is a sentence that is true (T) or false (F) but not both. The symbol denotes not, denotes and, and denotes or. If
More informationPigeonhole Principle and Ramsey Theory
Pigeonhole Principle and Ramsey Theory The Pigeonhole Principle (PP) has often been termed as one of the most fundamental principles in combinatorics. The familiar statement is that if we have n pigeonholes
More informationFOUNDATIONS & PROOF LECTURE NOTES by Dr Lynne Walling
FOUNDATIONS & PROOF LECTURE NOTES by Dr Lynne Walling Note: You are expected to spend 3-4 hours per week working on this course outside of the lectures and tutorials. In this time you are expected to review
More information1. Determine (with proof) the number of ordered triples (A 1, A 2, A 3 ) of sets which satisfy
UT Putnam Prep Problems, Oct 19 2016 I was very pleased that, between the whole gang of you, you solved almost every problem this week! Let me add a few comments here. 1. Determine (with proof) the number
More informationLecture 3: Sizes of Infinity
Math/CS 20: Intro. to Math Professor: Padraic Bartlett Lecture 3: Sizes of Infinity Week 2 UCSB 204 Sizes of Infinity On one hand, we know that the real numbers contain more elements than the rational
More informationWeek Some Warm-up Questions
1 Some Warm-up Questions Week 1-2 Abstraction: The process going from specific cases to general problem. Proof: A sequence of arguments to show certain conclusion to be true. If... then... : The part after
More informationPrinciples of Real Analysis I Fall I. The Real Number System
21-355 Principles of Real Analysis I Fall 2004 I. The Real Number System The main goal of this course is to develop the theory of real-valued functions of one real variable in a systematic and rigorous
More informationHomological Methods in Commutative Algebra
Homological Methods in Commutative Algebra Olivier Haution Ludwig-Maximilians-Universität München Sommersemester 2017 1 Contents Chapter 1. Associated primes 3 1. Support of a module 3 2. Associated primes
More informationChapter 1. Sets and Numbers
Chapter 1. Sets and Numbers 1. Sets A set is considered to be a collection of objects (elements). If A is a set and x is an element of the set A, we say x is a member of A or x belongs to A, and we write
More informationWe have been going places in the car of calculus for years, but this analysis course is about how the car actually works.
Analysis I We have been going places in the car of calculus for years, but this analysis course is about how the car actually works. Copier s Message These notes may contain errors. In fact, they almost
More informationInitial Ordinals. Proposition 57 For every ordinal α there is an initial ordinal κ such that κ α and α κ.
Initial Ordinals We now return to ordinals in general and use them to give a more precise meaning to the notion of a cardinal. First we make some observations. Note that if there is an ordinal with a certain
More informationBasic set-theoretic techniques in logic Part III, Transfinite recursion and induction
Basic set-theoretic techniques in logic Part III, Transfinite recursion and induction Benedikt Löwe Universiteit van Amsterdam Grzegorz Plebanek Uniwersytet Wroc lawski ESSLLI 2011, Ljubljana, Slovenia
More information1. Continuous Functions between Euclidean spaces
Math 441 Topology Fall 2012 Metric Spaces by John M. Lee This handout should be read between Chapters 1 and 2 of the text. It incorporates material from notes originally prepared by Steve Mitchell and
More informationNOTES ON DIOPHANTINE APPROXIMATION
NOTES ON DIOPHANTINE APPROXIMATION Jan-Hendrik Evertse January 29, 200 9 p-adic Numbers Literature: N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edition, Graduate Texts in Mathematics
More informationSeminaar Abstrakte Wiskunde Seminar in Abstract Mathematics Lecture notes in progress (27 March 2010)
http://math.sun.ac.za/amsc/sam Seminaar Abstrakte Wiskunde Seminar in Abstract Mathematics 2009-2010 Lecture notes in progress (27 March 2010) Contents 2009 Semester I: Elements 5 1. Cartesian product
More informationBackground for Discrete Mathematics
Background for Discrete Mathematics Huck Bennett Northwestern University These notes give a terse summary of basic notation and definitions related to three topics in discrete mathematics: logic, sets,
More informationSets. A set is a collection of objects without repeats. The size or cardinality of a set S is denoted S and is the number of elements in the set.
Sets A set is a collection of objects without repeats. The size or cardinality of a set S is denoted S and is the number of elements in the set. If A and B are sets, then the set of ordered pairs each
More informationFinite and Infinite Sets
Chapter 9 Finite and Infinite Sets 9. Finite Sets Preview Activity (Equivalent Sets, Part ). Let A and B be sets and let f be a function from A to B..f W A! B/. Carefully complete each of the following
More informationTheorem. For every positive integer n, the sum of the positive integers from 1 to n is n(n+1)
Week 1: Logic Lecture 1, 8/1 (Sections 1.1 and 1.3) Examples of theorems and proofs Theorem (Pythagoras). Let ABC be a right triangle, with legs of lengths a and b, and hypotenuse of length c. Then a +
More informationChapter 1. Sets and Mappings
Chapter 1. Sets and Mappings 1. Sets A set is considered to be a collection of objects (elements). If A is a set and x is an element of the set A, we say x is a member of A or x belongs to A, and we write
More informationA GENERAL THEORY OF ZERO-DIVISOR GRAPHS OVER A COMMUTATIVE RING. David F. Anderson and Elizabeth F. Lewis
International Electronic Journal of Algebra Volume 20 (2016) 111-135 A GENERAL HEORY OF ZERO-DIVISOR GRAPHS OVER A COMMUAIVE RING David F. Anderson and Elizabeth F. Lewis Received: 28 April 2016 Communicated
More informationRegular Languages and Finite Automata
Regular Languages and Finite Automata Theorem: Every regular language is accepted by some finite automaton. Proof: We proceed by induction on the (length of/structure of) the description of the regular
More informationFormal power series rings, inverse limits, and I-adic completions of rings
Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely
More information