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# MATH REASONING MATH 310

UW

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This 6 page Class Notes was uploaded by Addison Beer on Wednesday September 9, 2015. The Class Notes belongs to MATH 310 at University of Washington taught by Alexandra Nichifor in Fall. Since its upload, it has received 8 views. For similar materials see /class/192055/math-310-university-of-washington in Mathematics (M) at University of Washington.

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Date Created: 09/09/15

Math 310 Fall 2008 Midterm Information amp Topics Midterm Wednesday October 29m in class 50 minutes You may bring a 85xll sheet ofnotes onesided or twosided Test covers Chapters 19 of the text Review sessions in class Friday amp Monday Also outofclass optional QampA Tuesday time amp room TBA Study Sections 19 of text including examples and proofs class notes and handouts homework problems both collected and not collected Bring questions to reviews or office hours Main topics per section 1 The Language of Mathematics Statements Connectives and or not Implications Direct Proofs and proof by cases Proof by contradiction Induction skip 54 Strong Induction Sets elements ways to de nenotation subsets empty set operations on sets union intersection difference power set of a set complement of a set Thm 634 results and how to prove them Know how to prove a set is a subset of another or that two sets are equal Understand difference between an element and a subset Quanti ers universal and existential Understand what they are and how to use them Understand combinations of more than one quanti er and negations of such How to prove and disprove statements involving quanti ers Cartesian product de nition and how to use it Functions De nitions amp understand functions domain codomain image graph composition of functions Be able to come up with examples Be able to formally prove that the limit of a sequence is or is not equal to zero Functions Injections Surjections and Bijections De nitions amp understand when a function is injective surj ective bijective and invertible Be able to give examples Be able to prove injectivitysurjectivity for speci c functions Inverse of a function what it is and how to nd it You can skip sections 93 and 94 9959 gt1 9 gt0 Also understand inequalities and proofs involving inequalities Midterm questions may include computational questions for instance truth tables or function compositions definitions multiple choice or YesNo questions short questions like give the converse of the following statement nding errors in an argument short proofs eg set equality function inj ectivity one serious proof which may require direct proof proof by contradiction or proof by induction Math 310 Spring 2008 Final Topics amp Overview Final Exam 0 Section A Monday June 9 8301020 LOW 113 0 Section B Wednesday June 11 230420 LOW 113 You may bring two 85xll sheets of notes either one or twosided but stapled together and a calculator The nal is comprehensive covering chapters ll4 and 1920 of the text The sections which were not covered by the midterm will get more weight Review sessions in class Friday and optionally Friday 2304 in LOW 101 Study The text including examples and proofs class notes the endofchapter problems and the collected homework problems More practice problems are posted partial solutions will be posted later Bring questions to reviews or office hours Final exam questions may include computational questions for instance counting how many ways you can do a certain task definitions multiple choice or YesNo questions short questions like give the converse of the following statement finding errors in an argument short proofs a couple serious proofs Main topics per section I Recall the old midterm material The Language of Mathematics Statements Connectives and or not Implications Direct Proofs and proof by cases Proof by contradiction Induction Sets elements ways to definenotation subsets empty set operations on sets union intersection difference power set of a set complement of a set Thm 634 results and how to prove them Know how to prove a set is a subset of another or that two sets are equal Understand difference between an element and a subset Quantifiers universal and existential Understand what they are and how to use them Understand combinations of more than one quantifrer and negations of such How to prove and disprove statements involving quantifrers Cartesian product what is it 9959 gt1 8 Functions Defrnitions amp understand functions domain codomain image graph composition of functions Be able to come up with examples How to formally prove that the limit of a sequence is or is not equal to a number 9 Functions Injections Surjections and Bijections Understand be able to prove or disprove be able to give examples Inverse of a function what it is its domaincodomain and how to find it O N 4 The new material after the midterm Functions on subsets 7 and 7 Counting Finite Sets What is the cardinality of a nite set formal and informal definition The addition and multiplication principles proofs and applications Be able to use the inclusionexclusion principle Properties of Finite Sets Comparing cardinalities via injections surjections or bijections applying Corollary 1111 the Pigeonhole Principle Prop 1114 Corollary 1115 Thm 1116 1117 Sets with or without minimum or maximum elements Greatest Common Divisor what is it amp how to find it Combinatorics Counting Functions and Subsets Know the formulas for o lFunXYl o InjXYl o lBijXYl o PXl o of subsets ofk elements in a set ofn elements For each formula understand when to apply it Order matters With replacement What if you have multiple situations Binomial coefficients amp their properties understand prove be able to use Binomial Theorem understand prove be able to use Number Systems 0 Rational Numbers what they are showing that and of fractions are well defined proving a given number is or is not rational 0 Real Numbers understand infinite decimals both concrete ones and in proofs Counting Infinite Sets Which sets of numbers are countable Which are uncountable How do you prove it Be able to determine the cardinality of infinite sets by comparing them to known sets How many in nite cardinalities are there Why Congruences Understand congruences of integers definition and examples and relationship to remainders Know the properties in Prop 1912 and 1913 and how to prove them Applications such as the problems from hwk 7 Be able to solve linear congruences like the examples from the lecture handout on congruences Math 310B Fall 2008 Final Topics amp Overview Final Exam Monday December 8 230 420 MEB 235 You may bring two 85xll sheets of notes either one or twosided but stapled together and a calculator The nal is comprehensive including chapters 114 1920 and 24 of the text The material which was not covered by the midterm will get somewhat more weight Study Your class notes the text examples proofs assigned endofchapter problems and the collected homework problems More practice problems are posted partial solutions will be posted later Bring questions to reviews or of ce hours Final exam questions may include computational questions for instance counting how many ways you can do a certain task de nitions multiple choice or YesNo questions short questions like give the converse of the following statement nding errors in an argument short proofs a couple serious proofs Main topics per section I Recall the old midterm material The Language of Mathematics Statements Connectives and or not Implications Direct Proofs and proof by cases Proof by contradiction Induction Sets elements ways to de nenotation subsets empty set operations on sets union intersection difference power set of a set complement of a set Thm 634 results and how to prove them Know how to prove a set is a subset of another or that two sets are equal Understand difference between an element and a subset 7 Quanti ers universal and existential Understand what they are and how to use them Understand combinations of more than one quanti er and negations of such How to prove and disprove statements involving quanti ers Cartesian product what is it 8 Functions De nitions amp understand functions domain codomain image graph composition of functions Be able to come up with examples How to formally prove that the limit of a sequence is or is not equal to a number Functions Injections Surjections and Bijections Understand be able to prove or disprove be able to give examples Inverse of a function what it is its domaincodomain and how to nd it 9959 gt0 You may skip Functions on subsets 7 and 7 N N The new material after the midterm Counting Finite Sets What is the cardinality of a nite set formal and informal definition The addition and multiplication principles proofs and applications Be able to use the inclusionexclusion principle Properties of Finite Sets Comparing cardinalities via injections surjections or bijections applying Corollary 1111 the Pigeonhole Principle Prop 1114 Corollary 1115 Thm 1116 1117 Sets with or without minimum or maximum elements Greatest Common Divisor what is it amp how to find it Combinatorics Counting Functions and Subsets Know the formulas for o lFunXYl o InjXYl o lBijXYl o PXl o of subsets of k elements in a set of n elements For each formula understand when to apply it Order matters With replacement What if you have multiple situations Binomial coefficients amp their properties understand prove be able to use Binomial Theorem understand prove be able to use Number Systems 0 Rational Numbers what they are showing that and of fractions are welldefined proving a given number is or is not rational 0 Real Numbers understand infinite decimal representations both concrete ones and using abstract ones in proofs Counting Infinite Sets Which sets of numbers are countable Which are uncountable How do you prove it Be able to determine the cardinality of infinite sets by comparing them to known sets How many in nite cardinalities are there Why Congruences Understand congruences of integers definition and examples and relationship to remainders Know the properties in Prop 1912 and 1913 and how to prove them Applications such as the problems from hwk 8 Be able to solve linear congruences like the examples from the lecture handout on congruences or your homework Be able to apply Fermat s Little Theorem and Wilson s Theorem to solve questions like those in hwk 8 From class notes and handouts Euclidean Algorithm prime numbers etc

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