### Discrete Mathematics Rosen Section 2 1

all right questions I graded that little awesome ten exams in our futures I’m only spinning hopefully I mean use about six percent is a problem I was looking at look at your work we’ll see why people think that’s pretty short but eighty five exams that’ll put me in about eight plus hours worth of grading what I expect I mean good and bad I mean that’s that’s the hard part about a lot of this this isn’t one of those things where you can fake it we either get it or don’t then when you have a problem you either know it or not it’s just how it is the hard part is grading a fake it it’s where you I didn’t know what’s going on and I’m going to write a lot of words but then I have to prove I have to show it that right like well I didn’t connect that that doesn’t make sense whereas it was like a lot of words but you just made a small reasoning mistake finding the difference this is also where I get to introduce the next part it which is there’ll be a thing call extra credit proves okay starting before the next exam what I’m gonna do is I’ll block out a certain amount of proofs that I consider fundamental for the course and what you can do is at any time during the semester and block like you know for a few of them it’ll be before the next exam and then the for the third exam before the fourth exam three broth blocks of proofs and we’ll give the hand this out next class clock one is will do and I’ll explain this exam two – it’s going to be do before exam 3 and then block three is going to be do before exam four and what I mean by do this is to do an extra-credit fruit one of them will be the square to either a tional so if you already done it you can miss me and you can do it and get your extra credit point and if you look at the syllabus this is you get them all it adds like plus five to like the overall grade which is like a flutter great type of thing okay what I mean by do is you’ve gotta schedule times haven’t shift your office and then I can do sheet of paper what skirt you want it can you do it and then I look at it and I’m like no that’s not right and then come back another time try it again right and it’s either all right never sent wrong you you do it right there if it’s wrong I’ll explain why it’s wrong explain what you’re doing wrong and then you leave and you think about it for a while and you come back and you give it another shot you can to just choose to do this or not it’s just extra credit this is also probably ideally if we had small class sizes this is what your exams would be right where we wouldn’t give ABCD enough it’s like you learn discrete math right if you didn’t know discrete math I would sit there and point out this is what you cannot do you need to work this on this and this will go come back and do it we wouldn’t give grades good be oh end of the semester you need not to do this but that citizen that’s kind of funny we call ourselves educators for more letter givers it’s like oh look here be a CNA right so in terms of do you just simply stop by my office and do it you know here every day so three times or basically you know we have an hour after class between this one and the next one and then on Monday Wednesday Fridays you know after 11:30 no we work

on computers okay now on that I mean my other part of my job but you know that we can always fit in you know things like this this is also why I do it in blocks because a lot of people are like oh look there’s extra credit and they don’t do it and then we can finals I want to do it right here I want to do all of them and it’s just like yeah that’s just not even possible for me to do that and so it has to be scheduled out over time and so if you pass that point and like at the end of semester well I really needed to do it the answer is sorry you could have done it right so there’s a little bit of emphasis on you some time before exam to know before you can do them all tomorrow I mean yeah but some of them here all you have to do after exam 3 because the proofs are not going to be here at the beginning you know they’ll be proofs of number 30 like what’s the fundamental theorem you have prove the fundamental theorem of arithmetic which is gonna require induction and other tools that we won’t learn until later and most of these were a portion of these will be on exams anyways so if you study for them and like hey look I’m just gonna verify that I know how to do this and I’m gonna and these are also great it a little bit more detailed because I’m going to ask you questions now like I memorized this what about that what does that mean do you understand that the answer is I don’t know all right that doesn’t count I want you to know what these are well then these are gonna be so all right I started off at the beginning of semester talking about that this class is kind of a you know I like to talk about toys and rules in this class is a constant reboot that’s building on all things right so the very first block of math was the objects that we dealt with were propositions and the rules were everything that we could do with propositions when others saying we put them together how we make compound propositions other ways of interpreting them and then what our applications write again everything up to exam one could have could be if we wanted to done in more detail and they get an entire semester course alright but that’s that now that we have it it’s a tool we’re going to say okay this is an ability that we all have we all understand that form of math just like we all understand calculus and algebra trig in in terms of applications now we’re going to go on and make a new one and talk about a new type of math and starting on Chapter two we’re going to move on to set theory so this is our new math the idea it’s new maths so really what I mean by new math is we are talking about new objects with new rules number hold toys new toys and new rules and now the toys the rules that we’re going to be building up are going to be using logic especially when we do something as seemingly straightforward as set theory and we’re going to take it so today we’re going to talk about the objects themselves what are we going to be studying and then we’ll do a little bit of operations on them but I’m like logic we introduced our objects propositions and we had to wait a while so we could talk about the comparisons what does it mean to be logically equivalent what sets the way that they’re designed we can talk about the objects and immediately talk about comparisons before we have to go on to how do they interact with one another in terms of combinations of all sets and the new sets and other properties of it so today we’re going to actually define what’s our objects how we compare our objects how do we represent our objects and then eventually we’ll get into operations and applications of them all right like everything in math you know this idea of new is kind of arbitrary because these things come about for a reason propositions were you know this declares something to be true or false we fundamentally understand what it means when we did rules of inference right these tautologies these things are true why well not only can I show it and model it and make these little true/false tables and everything else and do this work I knew it because they had applications that came into being to model things we modeled discussions we

modeled reasoning and from that we knew already that it worked right because we had discussions before we formalized it and people could say you know I know Mark is telling me the truth I know mark is lying we knew that before we formalized it so we might modeling the same thing is here right we have a reason for this and so we’re gonna talk about sets now in particular this is for this definition this is actually gonna be naive set theory rather than just simply set theory which would be act true set theory is normally axiom axiomatic set theory going from steel naive because it’s easier to deal with but for you it said a set we’ve dealt with them before you know if we’re gonna have a whole branch of mathematics on these objects what would you think a set is we have grouping we have collection we have a wording of a bunch of individual objects right like I have a set of chairs I have a set of desks I have a set of students we have a set of buildings on campus right we know what they aren’t fundamentally I mean that’s the thing we’ve been dealing with them and now we’re going to try to happen okay they are we have sets so we need to go ahead and write down what they are and well actually at the very end of the second semester get to an issue where mathematically normally what we try to do we know things are we’re working with them now we try to write it down so we can formalize it intellectually – and model it right will actually run into things that we can’t maybe ever but maybe just not now define certain things when we get into computational approach no modeling will actually get to things that we’ll try to actually give it a formal definition but we can’t come up with one and well it’s kind of loose and so we won’t have a theory when we get to that you’ll actually have something look it’s looser than that yeah just we’re kind of proposing what things are but for us set theory we’ll have a good definition one of the things I’d like to say is it’s unordered because when we talk about a set of buildings on campus even the order that I name the buildings are really matters now if I say there’s a set of chairs in this room is there an order to well I could type an order on the chairs but when I talk about the word set I can stack them all in a corner and I really wouldn’t care if I put them all in desks I really wouldn’t care it’s just a bunch of chairs and so this is an unordered collection of objects normally the objects have a better name in the folk elements and there’s reason why because I’m using this definition of a set an unordered collection of stuff an ordered collection of elements but this is called naive set theory because this definition of a unordered collection of objects can actually lead to impossibility and the easiest example is the barber the male barber who will only give a shave to a person who has never changed them so right and so you would have okay I’m gonna divide up the world into two sets for guys right all right the guys at this barber can shave and that the guys this barber cannot shave and so we just simply split the world up have you ever shaved yes art you can’t you can’t go to him have you ever stated no all right you can go to him and the reason why this is naive set theory is where does he go if he’s never shaved then he’s in the group he’s allowed to do but that means he has to shave himself but he’s not allowed to shave himself because that’s the rule you know a lot of – so he has to go to the other group so which one is he in and his paradox if he’s in the first group he must be in the second but in the second group he must be in the first and so we just have that impossibility so if that’s your definition of sets unordered collections of elements you can easily get to kind of some things that have paradoxes that occur so what we do is a typical mathematical approach is really reasons why I call its naive that simple definition can lead to problems and so what we rather do is if you ever get a problem skip such things so now you see on theory does not deal with such things so you can simply say oh look this could lead to a problem we won’t talk about that one we try to put them off into special cases axiomatic set theory tries to create axioms at the beginning so if those become impossibility that we don’t have to worry about how they exist or not but naive is good enough for most of

things that we deal with some terms we can say things like a set contains its elements those we can actually have as a notation yeah element little a is is inset say capital a this is typically what we do use lowercase for elements of the case for sets we use the following notation little na is an element of set a if element a is not in set a we write it this way little a is not in that day and that’s the element of symbol kind of looks like a meanie counts like an epsilon it’s not it’s the element symbol that’s the element of all right so we have loads of reading capital letters for sets little letters for elements we talked about containing sets themselves how do we represent represent them all right so our new form of math sets themselves are the toys that’s what we’re going to play with and so we have some terms for them we have a little bit of notation for them but if I’m going to be working with sets I’m going to need to be able to represent them the first one is called a roster or list pretty easy the idea of representing assess with a roster or a list is to simply say okay here set a it’s made up of a square a triangle a happy phase the number 1 3 plus 2i the complex number and then and about the capital letter a and then mark what can be incest stuff right anything you want okay what can you put in a set you can put in objects of that our geometric you can put in numbers you could put in people you could put in sets right this could hear me actually you know the kind of self recursive problem what’s in a right maybe it’s a different set of a particular type you can really put whatever you want in here sometimes it’ll look funny say what am I going to do well I’m gonna put the set of the numbers 1 2 & 3 and then I’m gonna put in the number 2 and then I’m gonna put the set of negative 1 negative 2 and negative 3 my 3 to keep looking like okay if I look at this I actually have 1 2 3 objects in there right two of my objects happen to be sets one of the objects happens to be a number curly bracket just represents I mean if everything in here is going to represent a list of stuff what can I put in there anything you want the book will take advantage of that and make you look at really weird by making sets of sets of sets of sets you know have like all these nested parentheses and then you’ve got to figure out okay who’s on the inside an element is not right but in the end it really doesn’t matter what we put in it when we do roster methods a lot of times we’ll have say set s is made up of say 1 2 3 ellipsis 10 right if you don’t want to write all of the elements you can use the ellipsis to say do you catch the pattern I’m going to have 10 objects here I’m just not going to write all 10 please be careful I’ve had students who say I see 4 I see 5 elements in one of the elements is dot dot dot no I meant ellipses right because it is

a symbol all right you know and that’s strictly speaking you could read that that literally and then the person would be right darn but if we look at that the ideas do you see the pattern and you can put an end to it you can have say the set s and say negative 2 and then get a blonde 0 1 2 right all of the integers you know things of that nature and so roster method requires either you list all of them out or have a rule that’s in there that you don’t which that’s what the ellipsis means right the ellipsis is we all understand the rule follow the pattern right the rule is just by observation that’s the only time that you will be allowed to put that in is when that rule is obvious all right sometimes that’s not good enough it’s hard to be able to write things that have a lot of elements and let’s say that you look at the rule the rules a lot more complicated than you could possibly figure out say for example like irrational numbers then we move on to set builder notation set-builder notation is set ass is made up of okay we write the beginning of the set symbol but now what right now is a object variable a bar which means such that so it’s an object variable bar which actually means the words such that and on this side is a propositional function on object B what I mean all right say for example let’s say said s is made of all X such that X is an integer and X is greater than or equal to 1 yet less than 11 I could have written this in roster form right in roster form this would’ve been rather easy that’s 1 2 3 4 5 6 7 8 9 and 10 that’s in roster right what’s the advantage to set builder notation the advantage to set builder notation is that you can build complicated elements the question is when is it in the set when the propositional function evaluates the true things that are not in the set the propositional function evaluates to false so now we can actually get to for example the rational numbers there are going to be say they’re usually represented as a double bar q and these are all a over B such that a B are integers and B is not 0 and maybe have no common factors there work out of that so everything that we wrote before you know that was just simply a propositional function if those three things are true that’s a rational number if those one of those is not it is not a rational number and so now I can actually have the set the hard thing is like could you imagine trying to write the rational numbers in the list okay okay actually we will get to a thing that we will actually show that you can and there is a pattern to it the hard part is finding such a thing the other way that we can represent these besides as a list or set builder

notation we could graphically do it and the great way we would graphically do it as a Venn diagram all right for the Venn diagram we have a big square that’s absolutely required the big square has a u label on it all right what’s the you represent the u represents the universal set really it’s the universe of discourse think of set builder notation what are all the possible things they could go into your propositional function that’s your universe of discourse right it’s every element that we’re going to be possibly checking so we have to write a square I’ve had a lot of students that keep forgetting to put the big giant rectangle the Big John referred to that rectangle is the universe of discourse that’s who we’re talking about we have to have it and then you would then write circles yeah so a circle is a set normally you’ll label it like with the capital you can put that the label right beside it other times instead of putting the label right beside it you might break a small part of your circle and then put the label at that edge all right put an S right through there and the idea is that the line goes through it if you want to break it that way it’s your choice I tend to put it on the outside and elements are simply dots and so and again say that’s a lowercase and so capital labels little case for the elements and then really if you look at it it’s a visual roster method right these put dots for your elements in a circle that contains all of the elements and now what’s nice about the Venn diagrams is that you can see things where the propositional function has evaluated to true the elements on the inside and you also see the elements on the outside for the entire universe of discourse so it’s more of a visual representation of rosters without necessarily having to write all of the elements sometimes I’ll ask you to do that other times I want alright so we can write sets we can write elements we can do sets in a list we can do sets as builder notation we can do sets in a Venn diagram now that we have the basic ways of representing them there are some fundamental ones we should know we have number sets we have the natural numbers which is a double bar n is 0 1 2 3 etc up to never stopping so the natural numbers are the whole numbers plus 0 we have the integers which is a double bar Z which is all the integers we have a double bar z plus which is the whole numbers is the positive ends 1 2 3 so with natural numbers we have the integers we have the positive integers we already have our rationals which are going to be set builder notation these are all a over B such that a be puzzle write it this way a is a integer and B is a integer and B is not zero and I was saying no never go no knows what I means no common factors we have the reals real numbers right he knows in any particular form if I say

okay can I write this you know in some particular way and give a rule for it and kind of scratch your head and say can I think of her it’s like we kind of all know what real numbers are if you actually want to give it a descriptor of some sort you could say an R is representative all X such that X as a decimal form if use the words has a decimal form you can actually split off the reals in the sorry the rationals and the irrational to cross the decimals the rationals are the decimals that terminate or repeat the Irrational’s are the deaths decimals that don’t do that which means that they would not terminate and they would not repeat and so that allows us to sometimes have a natural way of splitting the concept of reals and this is kind of important so we could go over here to say we could have the rationals means decimal terminates or repeats now if we would look at that and the irrational is not rational what is not terminate or repeat let’s use the Morgan’s law it would not terminate and not repeat right and so that means the Irrational’s this would be a decimal that doesn’t terminate and doesn’t repeat I wanted to have examples of stuff like that say 1.01 or one point two two two repeating right both of the ones on the left if I could if we could actually even find it right we could represent as a integer over integer in simplest form with no common factors in the bottom is not zero everything like that on the other hand I can take this one say 1.0 100 100 one even though you see the pattern I can put the ellipsis because we can see the pattern does that pattern cause the decimal to stop which is an infinite number of zeros no does it ever repeat a block of numbers fixed no that means that the numbers on the left can be written as the integer of our integer in simplest form but the number on the right cannot kind of interesting and then we can go into the complex numbers which is a double bar see and complex numbers are simply equal to a plus bi such that a is a real number and B is a real number and I happen to be the square root of -1 right which is imaginary so there’s in the classic numbers that we need to understand so whenever we say things like real numbers complex numbers rational numbers positive ends and send the natural numbers we should know which ones we’re talking about double RN could also be called the non negatives right for the naturals other ones that we should know besides the number sets – the empty set empty or null set all right it’s always important to represent nothing I mean it used to be for a long time people think why do I need a representation for something that is not and it’s like zero is a modern concept it ends up being that now we actually need to have such things and so the empty set of the null set is a zero with the strong line through it and at what it represents is a set that has nothing in it it’s the collection of nothing what’s amazing is how many properties the collection with nothing actually has I shouldn’t be say it’s that surprising because like the number zero you know the number zero under arithmetic is the additive identity but it’s the multiplicative Dominator right you have all these things that it’s like when can

it be used when can it not be used and that same things with the empty set you know what’s interesting about the empty set we talked about comparisons the empty set ends up a set that’s actually inside of all sets and we would compare it to others the third would be the universal set u which is really this is actually the universe of discourse like what objects are you considering to be any being in your set the universe of discourse all possible elements of interest obviously there’s lots of universes of discourse so there’s a lots of universal set there’s not one universal set it’s like what elements you talking about are you talking about numbers then the universal set is all possible numbers are talked about students and the universal set is all students right they aren’t the same ideas possible ideas of collections the fourth thing would be singleton sets singleton sets as are simply all sets that have one element all right so those are all sets that we’re supposed to know so we’ve described our toys we can draw toys you can represent toys in here here we’re dealing with sets and now we’re going to say okay what are we going to do with them what are the operations but today we’re not going to talk about all the operations we’re going to simply talk about the first large class of them which is comparison and actually I to put plural because we’re going to talk about four types of comparisons we had logical equivalence these we had one right these are logically equivalent we’re gonna say set so we’re going to take we’re going to develop four different ways of comparing two sets so if I hold it set in my right hand and the set in my left hand there’s going to be four different things that I can talk about between those two sets all right the first thing I’m going to talk about is the idea of a subset all right since sets are collections of elements probably the most natural concept of a subset would be possibly would be represented as a Venn diagram and I would sit there and I would look at say here’s my universe of discourse here is my set say B and then here’s my set say a you know sometimes you look at elements right and we have some elements sitting around like this right and I look at this and I said you know what these are collections of elements and these you know easier to represent the set notation visually here with the Venn diagram to collect all those elements in a roster I noticed that all the elements that RNA happened to be in B which would say that to set a probably is another way of thinking of it is in B well how can I represent that visual representation and notation wise we’re going to say that a seems to be smaller right kind of like a smaller symbol how did you represent smaller and arithmetic like one number is smaller than another number less than or possibly if it was loose and I didn’t allow say less than or equal right and so I would say that a is smaller in a way how many’s the same sort of shape so it’s not a inequality symbol it’s a cup that represents the same idea right and or equal to B now what does it mean it says well I would like to say it looks like everything in a is unique how would you represent that

in logic so I would say a is a subset of B that’s what this is read as if and only if all elements for all elements that I see if that element is inside of a then that element has to be inside of me everything in a is in B that’s true all A’s are B’s that’s what I just said all the elements are B elements which means that if it’s in a then it’s in D they would okay with that being a definition seems to make sense no this symbol a is going to be called a subset if and only if that entire thing is true that’s what we’re saying I shall call to set a a subset of B only when all a elements are B elements and what if somebody says well I just noticed that all a elements are B elements then call a a subset of B that’s what we mean by ethanol and since that’s a totality that if and only if is actually a logical equivalency being a subset means literally all a elements are B elements so that’s a subset well the other thing would be if we would look at this particular problem like we had and let’s say that I knew like for this one now maybe you didn’t know that there’s actually some elements in B but you look at the badness I don’t really know if there’s elements that are in B but not a right you know any might actually grow to be maybe there may be this entire place is actually empty right and so we said maybe this entire thing is actually empty or not well we just simply say it’s a subset but on the other hand if you know if a is a subset of B and B has elements a doesn’t have then we say that this is a proper subset so we have subsets we have proper subsets well what’s a proper subset a is a proper subset of B what does that mean that means that okay for all elements if the element is na then the element is in B which is the same thing saying all A’s are B and you can find some other element pick something beside e you know X such that the X is in B and that X is not in it B is something a does not it’s bigger strictly bigger visually what that saying is there is literally an element here that’s in B and not in a then I would say it’s a proper subset if I do not know if that element exists I would say it’s at least two subsets subsets weaker it allowed a to actually be B all right so we have a subset we have proper subset a third would be the third type of thing would be equal what do you think it means for a to be equal to be same out everything in age and be everything in B’s and a right it’s the actual same quote-unquote set what does that look like logically that would say for all elements if it’s an A then it’s in B but if it’s in B it’s in a by

condition here’s a kind of a fun you’d also could note what’s another way of writing a by conditional left implies right and right and flies left but that would be subset notation that says that a equals B means that a is a subset of B and B is a subset of a if you want to use subset notation we could write it that way because the biconditional says a implies B means a subset B implies a means B is a subset so if you would show both those subsets you’ve actually showed equality odd features interesting things one the empty set is a subset of any set the empty set is a subset of all possible sets why do you think that’s true what’s the logical interpretation of subsets for all elements the element has to be in the left implies that the element must be in the right and this has to evaluate to true is the element in the empty set what’s in the empty set nothing so this is actually logically what for all elements false implies that the empty element is in s but what’s that that’s true vacuously so the empty set is inside of everything it’s pretty easy to see if we would look at it this way if I look at my universe of discourse and I write a circle here I keep trying to write a circle rather than just simply doing my circle out and let’s say I put my elements here dot dot dot dot dot dot dot dot dot can you always draw a circle in a circle that doesn’t contain any elements sure empty sets inside of everything you just drop certainly doesn’t contain any elements and the other sets has to be in there so it’s a subset of everything kind of an interesting feature like so what’s one of the properties of an empty set it’s a subset of all sets including itself a set is a subset of itself that’s just a trivial truth another one that’s kind of awkward here is this one one two three four four four and compare this to one two two two three three four four now look at those two and look at the definition of equality the definition quality says if an element is in a then it must be envied if an element is impede then it must be in a is the number one in the left does that imply that the number one is in the right yes is the number one in the left hand the right yes is the number two and the left and the right yes is the number three and the left in the right they’re equal what does equality check not multiple things of the same element just the existence of the element equality checks distinctness right it simply says is there a one well I have fun I have two of them is there a one that’s what I asked is one in the set I don’t care if you have a hundred or you have one of them do you have it the answer is yes or no and so we do not care about multiplicity and so because of that that allows for a question of problems of four students get come it caught up on is the idea of dinked so a lot of times because of this issue since multiples don’t matter on equality so usually we will stay with

distinct elements in other words if you have multiples of something don’t write it so normally we would rather simply say okay so when I talk about distinct I would say things like okay 1 2 3 4 versus the above those are the distinct elements so most the time when we write sets we’re going to write elements as they are in a distinct way rather than a non distinct way so we have ways to compare we can talk about subsets of proper subsets equality and Mosel say this last one just to do it B is cardinality which is equal to the number of distinct elements count the distinct elements we’ll call that the cardinality so the cardinality of one two and three with five is four right I see four distinct elements in cardinality we write it as a absolute value symbol but it’s actually a bar symbols as how many distinct elements that you see so just think this this matters to us so we have four ways to compare cardinalities subsets proper subset equality and next time we’ll new operations all right for attendance next class two point one number thirteen

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