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# Class Note for MATH 300 at UMass(2)

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This 3 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Massachusetts taught by a professor in Fall. Since its upload, it has received 17 views.

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Date Created: 02/06/15

MATH 300 FUNDAMENTAL CONCEPTS OF MATHEMATICS SYLLABUS Credits 3 Credits Note Math 300 students are required to register also for the lcredit co seminar Math 391A which is taught by an undergraduate TA Students will be assigned a seminar time during the rst week of class Prerequisites Math 131 and 132 Co requisite Math 2335 Suggested Texts 0 Class notes by Professor Hajir available upon request Class notes by Professor Meeks available upon request 0 The Mathematical Method A Transition to Advanced Mathematics by M Eisenberg Prentice Hall 1996 How to Read and Do Proofs by D Solow Fourth Edition Wiley 2005 0 An Introduction to Mathematical Thinking by W Gilbert and S Vanstone Pearson PrenticeHall 2005 General Course Description Math 300 is designed to help students make the transition from calculus courses to the more theoretical juniorsenior level mathematics courses The goal of this course is to help students learn the language of rigorous mathematics as structured by de nitions axioms and theorems Students will be trained in how to read understand devise and communicate proofs of mathematical statements A number of proof techniques contrapositive contradiction and induction will be emphasized The material includes set theory Cantor7s notion of size for sets and gradations of in nity maps between sets equivalence relations partitions of sets and basic logic truth tables negation quanti ers The material also includes discussion of the integers rational numbers real numbers and complex numbers One or two topics chosen from number theory group theory and pointset topology will also be included according to the interests of the instructor I Basic Set Theory and Logic Required De nitions 1 Given sets A B de ne A N B A UB A X B 2 De ne what it means for f A A B to be a 11 function injective or an onto function surjective or a bijective function 3 De ne the image of a function f A A B 4 For a function f A gt B and C C B de ne the inverse image f 1C 5 lnverse map f l 6 Equivalence relation on a set 7 Countable set 8 Uncountable set 9 Partition of a set 10 The power set 73A of a set A 11 Q R C Statements 1 lnverse map f 1 exists and is unique ltgt f is bijective 2 Well ordering principle 3 Principle of mathematical induction 4 Cantorls theorem 5 Fundamental theorem of equivalence relations 6 Binomial Theorem recommended Representative problems to solve and proofs of theorems l The composition of injective functions is injective 2 The composition of surjective functions is surjective 3 Write down the truth table for p q gt p 4 Prove 221 16 MnTTD using the principle of mathematical induction 5 What is the power set of 123 6 Cantorls theorem there is no surjection f z A A 73A 7 Q is a countable set 8 R is an uncountable set 9 Suppose 2 R A R What is the image off What is f 1 71 4 10 If A B C are countable in nite sets then A U B U C is a countable set 11 Find the multiplicative inverse of the complex number 5 7239 II Basic Number Theory De nitions 1 Congruence modulo m 2 Construction of Zm 3 Greatest common divisor 4 Prime numbers Statements 1 Division Algorithm 2 Euclidean Algorithm 3 Factorization of integers into primes Representative problems to solve and proofs of theorems 1 Find the remainder of 3100 When dividing by 4 2 Calculate the gcd of 484 and 451 and Write it as a linear combination of 484 and 451 3 Show that 5 is not rational 4 How many positive divisors does 180 have 5 When does a fraction have a terminating decimal expansion in R III Basic Group Theory De nitions 1 A group G 2 A subgroup H C G 3 Order of an element 4 Abelian and cyclic groups A generator of a group 5 Group homomorphism 6 Center of a group 7 Kernel of a group homomorphism 8 lsomorphism Statements 1 A nite cyclic group is isomorphic to Zm for one choice of positive integer m Representative problems to solve and proofs of theorems 1 Cancellation laW in a group 2 Uniqueness of identity in a group 3 Uniqueness of inverses in a group 4 List the generators of Z3 or Z2 gtlt Z3 5 The intersection of subgroups is a subgroup 6 The center of a group is a subgroup 7 The image of a group homomorphism is a subgroup 8 The kernel of a group homomorphism is a subgroup 9 A group homomorphism is injective if and only if its kernel contains only the identity element 10 The composition of group homomorphisms is a group homomorphism IV Basic Topology De nitions 1 Metric space 2 The ball Brp in a metric space 3 Open set in a metric space 4 Topological space 5 Closed set in a topological space 6 Limit point of a subset of a topological space 7 The closure of a subset in a topological space 8 Connected topological space 9 Compact topological space 10 Finite intersection property for a collection of sets 11 Subspace topology 12 Continuous function between topological spaces Statements 1 De Morgan7s Laws 2 Leastupperbound property of R 3 HeineBorel theorem Representative problems to solve and proofs of theorems 1 In a metric space an open ball is an open set 2 In a metric space the nite intersection of open sets is an open set 3 In a metric space the union of any collection of open sets is an open set 4 A subset of a topological space is closed if and only if it contains all of its limit points 5 The intersection of a collection of closed sets is a closed set 6 A closed subset of a compact space is compact in the subspace topology 7 The composition of continuous functions is continuous 8 The continuous image of a connected space is connected in the subspace topology 9 The continuous image of a compact space is compact in the subspace topology

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