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Introduction to Higher Mathematics

by: Mr. Jayde Auer

Introduction to Higher Mathematics MATH 215

Marketplace > University of Idaho > Mathematics (M) > MATH 215 > Introduction to Higher Mathematics
Mr. Jayde Auer
GPA 3.97


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This 8 page Class Notes was uploaded by Mr. Jayde Auer on Friday October 23, 2015. The Class Notes belongs to MATH 215 at University of Idaho taught by Staff in Fall. Since its upload, it has received 6 views. For similar materials see /class/227759/math-215-university-of-idaho in Mathematics (M) at University of Idaho.

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Date Created: 10/23/15
lSEMINAR IN TOPOLOGY OF THE PLANE FALL 2006 1 THE JOY OF SETS Just as musicians must rst learn to play individual notes on their instruments budding mathematicians must rst become adept at working with sets This leads to our rst de nition De nition 11 A Let is a collection of objects We usually denote sets by capital letters and we can describe sets in a number of ways For example 0 A 127 is a set with three members or elements in it We use the notation 1 E A77 to denote that 1 is an element of A On the other hand 47 A o N 1 2 3 denotes the set of all positive integers and is known as the set of natural numbers In this case 47 E N but 7139 N o B x 6x5 7 27 433p 7139 0 is a set whose members are well de ned but dif cult to determine explicitly When converting this de nition to English you should read it as B is the set of all x such that 6x5 7 27 433p 7139 077 In all of these examples we use the symbol 77 to indicate that we are de ning something namely the sets A N and B respectively The set N shows up often as do lots of other special sets with their own symbols such as the empty set De nition 12 The empty set is the set with no elements and is denoted by 0 Some other good sets to know are 0 Z 0 1 71 2 72 the set of all integers 0 Q n E Z in E Z in 31 0 the set of rational numbers and o R the set of all real numbers Although these are some pretty interesting examples we rarely focus on just one or two particular sets Instead we often start with a few sets of interest and then mash them into what we want using the set operations de ned next In these de nitions A and B are arbitrary sets De nition 13 A is a subset of B denoted A C B if every element of A is an element of B In other words A C B means that z E B whenever z E A For example observe that N C Z C Q C R De nition 14 A and B are equal denoted A B if they have the same elements Thus A B if and only ifA C B and B C A 1 De nition 15 The union of A and B denoted A U B is the set consisting of elements of either A or B Here or77 is meant inclusively Thus AUBz A or xEB De nition 16 The intersection of A and B denoted A B is the set consisting of those elements belonging to both A and B Thus A Bz A and xEB De nition 17 The di erenee between sets A and B denoted A B is the set of elements in A which are not in B Thus ABx A and Many of these operations can be represented visually with Venn diagrams These are very useful for developing intuition but a Venn diagram is NOT a proof In any case you should do Problem 11 Let X be a universal set and let A B and C be subsets of X Draw Venn diagrams representing the following 1 X A 2 A B U C 3 XAWXB 4 A B U B A This operation actually has a name the resulting set is the symmetric di erenee of A and B denoted A A B From now on you will often see problems that are stated as if they were true Your rst job is to determine if the given statement is indeed true If it is true provide a proof otherwise you must provide a counterexample verifying that the statement is false Problem 12 For any set A Q C A Problem 13 If A C B then A C C B 0 Problem 14 A B BA Problem 15 A C B if and only if A U B B Problem 16 A B U C A B U A 0 Problem 17 A B A B A Problem 18 A U B 0 A U B A U 0 Problem 19 A U B A U B A Problem 110 Let X be a universal set with A C X and B C X Then XAUBXA XB The universal set X in question is usually understood if not given explicitly Sets like X A and X B occur frequently enough to have a name De nition 18 The complement of a set A denoted Ac is the set of elements in X which are not in A Thus Acz Sometimes we need to consider lots of sets at the same time For example suppose that A1 A2 An are sets where n is some natural number Then Aiz Ai foreach 239 132371 i1 UAiz Ai forsome 239 132371 11 More generally we can use an index set I to describe a family of sets Thus we might write B Alha to de ne the set E whose elements are the sets A for each 239 E I We could then consider the union UA x z E A for some 239 id and the intersection A m m E A for all 239 id Problem 111 c U A M 61 61 Problem 112 61 61 De nition 19 The power set of a set A denoted 73A is the set of all subsets of A Thus 73A B BQA Problem 113 How many elements are in 73A if A 123107 Problem 114 Let n E N How many elements are in 73A if A has n elements We usually take the notion of a set for granted but this is a good time to point out that problems arise if we aren7t careful The power set 73A of A is an example of a set whose elements are themselves sets and this might make you wonder about the possibility of a set A that belongs to itself To that end consider De nition 110 A set A is abnormal if A E A A is normal if it is not abnormal Now let N denote the set of all normal sets N A A is normal Problem 115 1 lfN is normal then N E N 2 If N is abnormal then N Z N 3 ls N normal or abnormal Mathematicians like to rule out irritations like this so most of them accept axioms the ZermeloiFraenkel axioms of set theory that prevent such things from popping up Although we wont worry about this any more you might want to investigate mathematical logic on your own if you want to learn more De nition 111 A nonempty set A is nite if there is a one to one correspondence between the elements of A and the elements of the set 1 2 n for some n E N The empty set is also nite De nition 112 A set A is in nite if it is not nite De nition 113 A set A is countably in nite if there is a one to one correspondence between the elements of A and the elements of N De nition 114 A set A is uncountable if it is neither nite nor countably in nite De nition 115 A set A is countable if it is either nite or countably in nite Problem 116 Show that Z is countable Problem 117 Show that Q is countable Problem 118 We discussed one way of enumerating the rationals in class Using this basic idea to enumerate the positive rationals how long do we have to wait for the positive rational number mn to appear in our list In other words determine a number N so that you can guarantee that this number will appear within the rst N terms of our resulting list of positive rationals Don7t try to be exact Just get a rough estimate for N Problem 119 ls R countable Some further comments on the set R are in order We de ned the rationals Q C R above the irrational numbers are those real numbers which are not rational For example Problem 120 x2 is irrational The reals are a well mixed blend of rationals and irrationals In fact the following hold 0 Suppose that a and b are real numbers with a lt b Then there is another real number c such that a lt c lt b 0 Suppose that a and b are rational numbers with a lt b Then there is an irrational number c such that a lt c lt b 0 Suppose that a and b are irrational numbers with a lt b Then there is a rational number c such that a lt c lt b 2 GETTING TO KNOW YOUR NEIGHBORHOOD De nition 21 The plane R2 is the set of all ordered pairs of real numbers R2 zy Ly E R More generally for a positive integer n E N we de ne R to be the set of all ordered nituples R p1x2pn z E R for alli De nition 22 If p 101102 and q 11412 are points in the plane then the distance between p and q is dltP7 I 3 101 i 102 102 Q22 More generally if p p1pn and q qlqn are points in R then the distance between p and q is 10139 i 102 M 611 Q i ll De nition 23 If p 101102 and q 11412 are points in the plane then the dot product of p and q denoted p q is the number 10413 10111 P2Q2 More generally if p p1 10 and q ql q are points in R then their dot product is the number 71 19 q ZPiQi i1 Observing that dpQ p7 Q p 7 1 leads to De nition 24 The norm or length ofp E R is the number lpl d107010 10 Problem 21 Prove the following results related to the distance function and the notion of length just de ned 1 For any two points pq E R dpq 2 0 Moreover dpq 0 if and only if p Q 2 Prove the Cauchy Schwarz inequality for points pq E R 19 q S W M Hint First compute p7 q p 7 q for points p and q such that lpl lql 1 Then for the general case start with arbitrary p and q and turn them into vectors of length one 3 Use the Cauchy Schwarz inequality to prove the triangle inequality for points 1991 E R lpql S W W Hint Start by computing p q p q 4 Use the previous version of the triangle inequality to prove its alternative formulation for points pqr E R dpQ S dpr 610 De nition 25 If po 6 R and 8 gt 0 then the sphericalueighborhood about p0 of radius 8 is the set Bp08 p E R dpp0 lt 8 To take a break from all of these de nitions do Problem 22 Sketch the following subset of the plane 327271 U 347271 U 33717 De nition 26 A set A C R is open if every point p E A has a spherical neighbor hood contained entirely in A In other words A is open if for each p E A there is some 8 gt 0 such that Bp8 C A Note The radius 8 gt 0 in the previous de nition will usually be different for different points p in the open set Al Problem 23 For any point p and any given 8 gt 0 the spherical neighborhood Bp 8 is an open set Problem 24 If A and B are open sets then both A U B and A B are open De nition 27 A set A C R is closed if its complement Ac is open Problem 25 If A and B are closed sets then both A U B and A B are closed Problem 26 Determine whether the following subsets of R2 are open closed both or neither 1 9621 957ygt0 9611 2yzlt4zy 2yzlt1 gay 9392 5 967296716Z 6 MI t W lt11 1 1 9577 3 E7 EZ 8 9611 2yzlt1U272 Problem 27 Let I be an index set and suppose that A is an open set for every 239 e I 1 UAZ is open 13961 2 HA is open ieI Problem 28 Let I be an index set and suppose that A is a closed set for every 239 E I What does the previous problem tell us about the union and intersection of these sets De nition 28 The point p is a cluster point for the set A if every spherical neigh borhood about p contains in nitely many elements of A De nition 29 The closure of the set A denoted A is the union of A and all of its cluster points De nition 210 The point p is an isolated point for the set A if there is a spherical neighborhood Bp 8 such that Bp8 A p De nition 211 The point p is a boundary point for the set A if every spherical neighborhood about p intersects both A and A0 The boundary of A denoted 3A is the set of all boundary points of A Problem 29 For each of the subsets of R2 de ned in Problem 26 determine the cluster points of the set the closure of the set the isolated points of the set and the boundary of the set Problem 210 A closed set contains all of its cluster points Problem 211 A set with in nitely many elements has at least one cluster point


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