Lecture 6 21 127 Concepts of Math 09 10 2012 Administrivia Homework 2 is due on Thursday Please write your Andrew ID on your homework It is how grades are entered into CMG so help us out There is now a place on the LaTeX template for this Reminder about tutoring options EXCEL Tutoring Work in groups get help with course material from a group tutor Very helpful Sign up in Cyert Hall B5 Walk in Tutoring Tutoring for 21 127 will be available on Wednesday nights from 8 30 11 00 p m in both the Donner Reading Room and the Mudge Reading Room As usual the service is free of charge and students can drop in for help without an appointment Sets Definitions and Examples Hopefully you spent some time over the weekend internalizing the concept of a set We ll remind you of our definition Definition A set is a collection of objects that have a common well defined property The objects contained in a set are called elements of the set The mathematical symbol represents the phrase is an element of and represents is not an element of Remember that the order of the elements is irrelevant as are repetitions of elements Set Builder Notation A very convenient and popular way of defining sets is using the set builder notation We call it this because we are building a set by drawing elements from a larger set of possibilities and only including those that have a particular property To do this we need to tell the reader what the larger set is and what the common property is Let s see a few examples S x N 1 x 100 1 2 3 100 T z Z we can find some k Z such that z 2k 4 2 0 2 4 n o U x R x2 2 0 2 2 V x N x2 2 0 The last two examples show how the context is extremely important The same common property satisfying x2 2 0 can be satisfied by different elements when we change the larger set from which we draw elements Two real numbers satisfy that property but no natural numbers satisfy it This is why it is absolutely essential to specify the larger set a definition like U x x2 2 0 is totally meaningless because it is ambiguous and could yield completely different interpretations We are really learning a new language here and these are some of the basic words and rules of grammar We ll need some practice translating these sentences to English in our heads and out loud and vice versa For example we can say the definition for S above as any of the following reasonably S is the set of all natural numbers x such that x is between 1 and 100 inclusive S is the set of all natural numbers between 1 and 100 inclusive 1 S is the set of all natural numbers x that satisfy the inequality 1 x 100 S is the set of natural numbers x with the property that 1 x 100 Notice that all of them identified the larger set and the common property the only differences between them are linguistic grammatical and they do not alter the mathematical meanings Here are some other examples For each of these let S be the set of students in this class and let C be the set of students at CMU SA x S student x s first name begins with A S2 x S student x is a sophomore C x C student x has passed 21 127 already Practice reading these sentences out loud and interpreting them The Empty Set Look back at the set V above What does mean Definition The empty set is the set which contains no elements It is denoted by the symbol Sometimes we will use so we will know what you mean if you write that Notice we said the set it is unique This is in fact an axiom of set theory the empty set exists Sets Elements and Subsets Definition and Notation Keep the bag analogy in mind An element of a set x A means we can reach into the bag A and pull out exactly the object x What follows is the notion of a subset this is a way of comparing two sets bags Definition Let A and B be sets If every element of A is also an element of B then we say A is a subset of B The mathematical symbol for subset is so we would write A B If we want to indicate that A is a subset of B but is also not equal to B we would write A B and say that A is a proper subset of B We can also write these relationships as B A or B A respectively In these cases we would say B is a superset of A or B is a proper superset of A respectively The standard sets of numbers we use follow some nice subset relationships N Z Q R C Set builder Notation Creates Subsets Look at what we did with set builder notation We identified a larger set a universal set and then drew out some of its elements to include in a new set In so doing we are guaranteeing that the new set is a subset of the larger set That is x S some particular fact about x holds true S holds true regardless of whatever that fact about x is 2 For example x R x2 2 0 R holds true and we observe that 2 2 R because comparing the two bags every element of the set on the left also appears as an element in the set on the right By that same logic then x N x2 2 0 N holds true That is N In fact this subset relationship holds true for every set Fact For every set S it is true that S That is there are no elements to draw from and compare to S so the statement given in the definition of subset is vacuously true That is there are no cases where it is false so it has to be true Hopefully we now see why the following statements are all true 1 N 1 N 1 6 N 1 N N Z N Z Set Equality We previously said that two sets are equal if they have the same elements The subset relationship allows us to express this property very nicely Definition We say two sets A and B are equal and write A B if A B and B A That is to say that A B we need to know that whenever we have an element of A it is also an element of B and vice versa In the future we will use this definition to prove two sets are equal Sets of Sets There is nothing to stop us from allowing the elements of a set to be sets themselves Here s an example A 1 Notice that A has two elements Each …