### Math 308 Lecture 1 – Introduction to sets

syllabi my name is John Smith and I will be your instructor for this summer this is your section and these are the meeting times and of course the room which is also which is pretty cool so this is my you know I will be asking you guys to contact me at this email shortly office hours by appointment and this is my website you should go there because the website for this classes page is also there for this particular class the textbooks you need two textbooks for this class most of our friends when you spend with the first one we’re going to do the first ten chapters from mathematical proofs a transition to advanced mathematics by charging and then we are going to do chapter one and maybe a little more time permits of tiny reference from elementary analysis that they are taught us by Ross so these are two textbooks the second one in particular you can actually download the pdf version of this textbook free if you’re on the campus right so once you’re on CC anyway network and you can just go to if you go through the math through a website official one for the class that have a link from which you can download the textbook so you can let’s go there a possibility you are here and you just click on the link and you can download the second texture the first step would not so we’re not so fortunate with that one and that one is sort of expensive it’s it’s definitely more than a hundred dollars I don’t know if it’s closer to hundred them however the the first textbook actually really like the textbook I think it’s a really good textbook not just as far as my textbooks go but textbooks in general I think is very well written and I bought mine the first edition was really expensive as well I don’t regret it I actually really liked the textbook so here you have a math department location doesn’t use bursaries need to go there at the math department website you should have those and I gave the website of the page and of course my website always any websites top there’s not a lot of class we’re not really important they won’t make a little too much annually the classes are not going to be very confrontational greatings the great scenes here in the table it’s pretty generously expected it’s already pretty generous I’m still getting that removed from the matter part of that is 40% of your grade final the other 60% is broken down here so we have 15% in fact so there will be written homework for each chapter what I will do is every class I will either collect your homework and grade so that the selected number of problems or I will give you a quiz that is based on homework so if you did the homework you should you find participation so actually because just for showing up right so even these blackboards here at 5% just bite me here so don’t throw that away in class tessai 95 my living – right there worth 40% energy so each one will work about 20% and the final is worth 40% extra credit I don’t have anything in mind what something might come up there are many hopes of them yes I don’t have I do take attendance I called rule in class earlier today so I’m arguing if you arrived after I caught your name and of course if you were going to get out sooner or later you should have least send me an email at all itself okay he’s within one day of your absence

or late this because if you put too many too many nonsenses but we’ll have to give you a failing grade so the Churchill and so a long time my expectations work at it don’t slack off expect you to work hard is the summer costume you really intense like three minutes okay it’s an intense course even during the regular semester so it’s very important like you know what’s going on so I gave you the syllabus even the topics and chocolate recovery and the homework also the card so you’ll have a fairly good idea of what is coming up I highly recommend that you read ahead and start having a great birthday rockin all the stuff all the topics for anger them in class I’ll be good so you can follow easier whilst I say here make sure you take number of definitions and computation and dumb read and do homework just the usual stuff yes so about this course has anyone ever team scores before yeah but if it wasn’t a discrete form yeah mom put that part here we’ll go up to a little bit of set theory and just some problems of the room okay so yeah especially the first three sections should be okay with you guys so but no one else around did you guys think groups in that class yeah it’s different some professors teach that very different something that a lot of them some complications there okay all right so some of you have some experience stuff that will be good the rest of the nuts and hopefully you can catch on quickly but the class is very different it’s not going to be very computationally I we won’t care too much about computing something or just working on a problem or something that they focus on this possibly more about argument about reading through something logic about how to think mathematically that’s sort of the subject matter itself isn’t too difficult usually like I mean will hardly do anything like calculus or anything that happens the level element that not that will do it won’t be hard but it’s just we’re going to approach it from a very different angle it’s not like I’m gonna do you can expression final derivative it’s not going to be like that it’s more it’s going to be I mean you’ll see when we get to it but it’s going to be very different it’s more of other things to a problem or how to know if something is true or not and how to actually convince someone else that it’s true or not give me an argument you know so it’s not like I could say this polynomial I mean I might ask you something like if you know function is differentiable any notes benteen’s give me an argument convincing you know it’s going to be like that or you’re going to live in write down an argument it’s a very writing intensive course which is probably weird for a lot of people that amount of course it’s like that when I was in my undergrad here there was a requirement that you have to do a certain number of writing horses right severe disease course path right certain amount was that he was like 3,500 or as per semester and you know so a lot of people use history courses philosophy courses I do math courses this course actually comes to the right so if you have such a requirement you can actually go to the math department and say you’ve a regularity this person took math your weight becomes as a riding horse it’s very writing intensive it’s not computational it’s about argument it’s about proof and so that says it’s going to be very different before pay attention to definitions is going to be really important most more so than ever that’s always really really important so whatever objects you talk about is whatever whatever talk about you know what it is even of what it is precisely because that will actually be costing a blog okay there are times when in my mouth pasta I’ve been confused like a whole semester or something it’s because I got

the word on the definition oh it’s set or so enough for every and completely routine is everything like you have to be really specific really detail-oriented really organized you just have to like know your stuff right sometimes we would have to find things you can’t find something if you don’t know what you’re looking for right you need to know if I say find this you need to know what I’m talking about so you don’t want to look work right or I say prove that this thing is this kind of product you need to know what it needs to be done about it you can’t do it otherwise right so it’s very fun that you read you take Nordic definitions you take a note of notation because as I said we’ll be doing a lot of cruise and a proof is really a way to make sure something is true for one and secondly is to convince other people that is true right so don’t just be like you know for hot oh I’ve discovered this amazing proof and you never tell anyone how to do it and then everyone’s all confused and no one knows if they can believe you’re right it’s very important that you be able to write down this is how I know what I say is true and you should be convinced by in this argument so it’s going to be begin to write something down clearly it’s going to be very important so it’s not like you know we show these steps of computations and it’s going to be fine you know you have to justify every step I know I can make this assumption because of this I know I can eat this step because of that and you have to literally write them writing English sentences most of the time right so it’s you’re going to be practicing writing skills so very different they’re going to be different skills you’re going to have to be more creative for a while so of course you guys have all pass a lot of math courses at least count three noise a year I would assume so you sort of know before this class or classes below this level which we call elementary mathematics vector calculus elementary mathematics that’s not where it’s computational where it’s not really about being creative insight ok differentiate this thing ok you have this room we just follow the road right like you only needed two skills to do well in math class before this right for mentality that’s all you recognize a certain form I know another one who goes with that form and apply it right we want to find them so polynomial right you know all about all the algorithms that go along with these formal in fact about foiling you technically the square right so with the quadratic comes with it all these algorithms and that’s all you have to know you know how to recognize it so these are quadratic the algorithms that go along with regret it and you’re good that’s all here it’s a little bit trickier you are going to have to talk about things and through things where there is no clear path where I’m going to throw some of the I am and you’re right there’s no sequence of steps for this kind of thing I have to sort of fit or something up I have all these theorems over here after somehow arrange them in some logical order that can apply to this problem in the strictest sense we won’t be doing anything new in this box I mean the stuff we’re improving the stuff that everybody knows it’s all reinventing the kill it’s not gonna be crazy but for you it might you that’s going to be a new child sort of idea of being creative and sort of figuring out how to attack a problem when there is no clear to any sequence of steps that is going to be one of the main challenges here so it won’t be like another class or area like it should use integration by parts like okay you memorize the method you do 20 problems and you’re good it’s not going to be like that a lot of problems will look very different even though we deal with the same thing and there is no like one-size-fits-all so that should be for this class sometimes that’s gonna be confused try not to do see me try not to be one of those people work I have to understand every single detail before I can move on it’s not gonna happen you’re gonna go disappoint yourself they’re going to fall behind by doing that sometimes you have get to the main idea you just probably write something is important enough for you to get that thing I’ll emphasize it but in general don’t be like the OCD would have you write something down the notation use that argument that’s fine but sometimes you’re not gonna understand everything the understanding will come don’t try to force it it will come and you stick to it I promise so sometimes you just need totally confused and there is another weird thing that will happen here more so than

in other complexes there are going to be times when you understand everything oh that’s fair go no I absolutely get it I went for point A to point B I follow yourself I fully understand it but if you never come up without myself if you can put that on a test that we’ve not been able to good so it’s sort of a weird thing that you’re going to be able to understand something and not understand it at the same time that’s going to happen don’t worry buddy is completely normal right so it’s like yeah let’s see what you did but there’s no way that comes as well I mean I experienced that every my professor you’ve ever had has experienced that point where we’re like okay yeah let’s see what you did you are smart enough just take your brain time to catch up there have been times where I was you know like after I pass this class and you know someone comes to the theater by the way also I would get too many venture so Cyr might come from this class of materials in her right there professor might keep on this crazy extra credit problem right in there like I don’t know what to do here you know and you know I read type I read you to follow me you know and just go through it and it’s sort of weird especially the first time this happens to you but you sort of have the idea that I know had I been 83 or a like wouldn’t have known what to do like there’s I just wrote something on the board it’s a correct solution but I’m actually surprised I came up with it comes back in theory and the CEO say the same thing to me I see where you live Devon but I never cross myself to silver hearing so it will come right so don’t be afraid to okay I cannot come back it will come if you stick with it eventually your your brain will catch up to you you just have to respect the attention to the things you should be aware of for problems like I said here I have a list of former problems already on the second page and I mentioned here is one assembly homework we just don’t have time to drive kids on the soccer field intestines I hope everyone here is how many meters we have what are the other engineering so much miners I guess anyone here just alright okay no problem with that so yeah basically be good to me hopefully you you’re not one of those struggling students because you will have a hard time keeping them so yeah when you hadn’t homework I do expected to have follow certain guidelines or I will deduct points of course really work very important that you’re able to write things down clearly here in this class so don’t make too far only two to two largely means that someone would not make sure you write so that you know your classmates will interview solution and then your style you’re not just writing for me you’re writing for everyone save your own money which consists of more than one pages properly label you over right now your name right on the homework number or the topic number so on this list you see the homework I actually don’t hurt the homeworks see the right now oh this is homework number one so what I am so let’s say you’re submitting the first time I got the top of your and I expect to see something like this you have romaine and one and then you have section 1 section 1 6 so that’s what we have the top of your homework if you have it takes several pieces we often will expect it to be stapled make

sure you do one set of homework so right so separate your goomers this might be rare in this class but let’s say I cover like two topics in one day and you have to submit over one and one or two on the next class expect needs to be separate don’t staple them together just separate them and them separate be needs write legibly and all that yes so I expect you to follow certain guidelines like I said be organized going to be very important in this class and so I want to practice being organized you’ll need to have organized sports can be able to organize a random array of information some need to practice that expect to remember all the math and done for this like I said most specifically use anything for my calculus or in your group lower classes but movies an array of things yes here I speak about contact by the end of today’s class by the end of today expect you to email me at the email address I provided on the first page so by the end of today I expect you to email me at the email address listed at the beginning of the syllabus the top from your course and the section I will be having into an email list as well as program that I can send out reports and other things to you so you can see primarily in the course as time goes on yep reader you know when I send it to you I said very important for me long time I don’t expect that’s out a lot of information or knowledge in this class because you don’t realize my class last semester they complain about them so much we don’t want to do anymore what they ever pay attention to anything I say they never read my emails and then they get a toke because of it and then they blame me so well what do you need we didn’t have to do that homework I spent like two hours last night when I could study for the quiz I said to email saying don’t do that homework expense study for your quiz it’s like yeah he attention attention to detail very important right very very very super important I cannot feedback encourage me back on my website which is also after I finish the topic right so yeah so I did it’s written everything but I didn’t mention it yeah so like after I finished chapter 1 the homer photographer one is different whether I remember to say it or not you should have a break right so I will make sure I actually announce when I tried it for chapter one we’re 6 the car team of the part that says so until I saw her that you don’t chop the one yes so homework is due the class after I finished the session the child right so feedback encourage feedback if something’s working for you or something is not working for you as far as my teacher in certain I expect he told me on my website there’s a place where you can give me an honest feedback and then the accident you may get constructive so don’t send the message like they’re teaching sauce you know like say you’re teaching sauce because right you don’t listen that would be better than this yeah so you can do that it’s anonymous so I don’t know what you think I encourage me like I won’t penalize you at all for some reason but I’ll do that it’s promised some feedback is like should not be anonymous I have kids all the time yeah I guess I’ll continue swimming there all right

so just yeah so be aware there’s certain things that shouldn’t be anonymous anybody help help it available we have a tutoring center here in the Marquita tall building on the fourth floor and for 180s on the arts of the summer are Mondays and Thursdays welcome to 5 p.m it tearing it opens I believe on Wednesday I also appeared there a lot so typically after this class you guys might grab lunch or something and then I’ll actually get over there it’ll seem either a lot of other tumors I also hear some websites that might be useful to check out so some services you might think of taking advantage encourage you to work with your classmates you can work together on one part just make sure you something your own homework and actually know how to do things yourself you don’t pass rules through cell phones keep them on silent eating and drinking your coffee that’s garbage don’t cheat don’t like cheaters next page is the academic culture though some things are important you might need to know about so I always include that in Silvis this is the homework list where you see the homework number the section of the topic and these are the problems that I expect you to do tests are not cumulative except where it’s incidental sometimes things just overlap but the final will be tainted right so the test I didn’t put a date because it’s really depends on how fast we cover the topic so they see the tests are positional so once I complete up to chapter 7 then I would maybe test one and then test to comes after from chapter 8 onwards to the end of the second book just to those for me team so each test will cover seven topics each map really sort of begins after test one night everything else is just like prerequisites you doesn’t need to know what these things are and how they operate so that you’ll be able to use them when we get to chapter 8 India so it’s gonna be a lot of preparation you think everything in test one is just preparing you to actually do things that you need to do as a mathematician yeah especially for those who you haven’t seen your proof course before you’re going to realize that math is a very different animal from what you think is most people when they think of algebra geometry is so much more tip of the iceberg we’re going to start going below the surface how the question you here that I’d like you just to see what pattern so you can just fill it up it’s a separate agency anonymous though you don’t have to write your name on it and you can just hand the or no I think that’s something to say are there any questions those occasionally can I have your names is rockin Kobe Bryant I’m not sure can actually borrow these

text books from the library probably time but I’ll try to get some pixels for like the tutor Center yeah give me now retained of cost this is are you guys actually encouraged to take this clockwise business advisor business like begin so they’re actually so I started recording my classes like a couple of years ago you know what it seemed to like it I’m not sure how much recording this class and if anyone else excites you guys because it’s one of those horses eyes of both standard and non-standard at the same time like there may be different ways you can teach it so some people like to emphasize keep the math simple and just emphasizes strategies and techniques for doing something although I really do want to emphasize you know how to approach something rather than just be this big that’s crazy hard mathematics because you have you know we start to do advanced compass which is what this course is preparing you for then you’re going to deal with some hard subject matter right and you’re going to need to know how to think about or to approach the problems there are also professors and elective right out like the truce for theories I rather not do that for the most part because pretty much be copying the textbook into that I like right on to them for a theorem or very important proof if I think it will be instructed in some way but in general I will be writing down falafel so I will be writing the definitions for everything and talk about say you know specifically what things to focus on I won’t be going through everything in the textbook from all 10 chapters so I’ll write out definitions and everything from one time so you can know what I’m doing and I encourage you to take notes right it’s dumb like I said knowing definitions super important so you should practice

writing them down and memorize we are what map really is about turns out everything we’ve been doing up to this point we just a special case what about incision really do you probably never seen before and you’re not making this course but let me tell you be able to you’re going to be able to start appreciating it and now we give the project but before logic we’re going to talk about sense because a lot of times logical expressions can be expressed in terms of set so let’s talk about sex for a while a lot of you should know about this stuff for the most part so today’s class is mostly going to I definition it’s nothing too crazy I don’t know if we didn’t finish the whole chapter we’re going to try so just to make sure that everyone’s on the same team we’re going to start from the beginning definition what is the sense right collection of moments right objects in the very renewed sense of the word can be numbers letters names whatever and these objects are called elements or members so that is what a set is sets are denoted my uppercase letters and so a etc wild Anders lowercase letters notation for members an element a belongs to set e we right listen is an elemental I can say that in any one of those phrases otherwise right so humble right state is the set containing this room numbers 183 I can say one is an element I can say for it’s not an element of it this is because the second one the element one is a member of the set while I set the number four is not important very important set though shows up all the time is called the empty set or sometimes called the null set denoted by just right squiggly brackets with nothing in them or like a zero a strikethrough this is a set of no elements literally an empty set so with your primitive meat but we don’t know how to write proof yet so I

won’t do that it’s actually one of mine it’s not that that’s actually talked about how to describe this there’s something called a Rustler method you don’t know what it’s called we need to recognize it when you see it promise so I already showed it here is a set a is that one I can say is three so that’s a very important competition here ellipsis I use to show our novice so sometimes you know one right on every element of us say because I want to write all the numbers from one to a hundred but you can start off from some pattern like you can see here that the number is increased by one and then you must but this is show that that same tired continues until we get to get right there is another there’s another way to use of their system you can say that element C is the number one – that’s probably custody today so I’ll see the zero three six leave the pattern panties forever referre it’s not very not nautical right so which input it looks at the engine study that you know they’ll it will continue going on forever with this title right so you know that after nine you’d have 12 and so on and so forth it’s very predictable you can use their system sort of cut down they want you to do is note the elements upset and you set themselves so I can in how to set say one to the empty set and the second three this is syrup so I can say that three which is a set is an element of another separate you cannot set continuing sex does not a strange thing at all no we’re going to talk about subsets but three is not it’s just an element of the set what would be a subset is part a 17-3 Susan the set containing the set three now didn’t define subsets we’re going to so if you don’t want to just don’t worry I’m going to tell you eventually but yes three is here an element but it is not a subset set

in the notation for doing this it’s called set notation again unicode not what the name of it is but I can see the set a is the set of all element axis such that it’s not P of X or equivalently some people I keeps pulling insert the hard feelings so P of X for example you can say you know that a Eva set see the sound of all excellent I also should mention about this this bar or the colon it’s read as such that so this is the set of all X such that this is true so for example I can write set of all X such that absolute I X is less than 3 where X is an integer if I need to flesh this out I need to right now this is specifically what the members of the set are what would they be so instead of writing down a list like this I can actually describe it as all elements that will fill some part and a lot of times writing this down is going to be a lot easier and a lot more efficient than actually twisting specifically what the members are so we will do that actually quite often describe us that in terms of some property that I expect the elements of all I’m actually going to define what it is if you don’t know so this would be the set that looks like zero plus or minus 2 plus or minus 4 these are called two even and we’re going to play around even and obviously there’s a lot I can say oh is the set of all X such that X is equal to 2 n plus 1 so the guys in here come on one two three spot this is called set of odd those are examples of how you can write down all the elements that you don’t want list anything to actually just right now somebody that fulfills a certain property and you can actually got a little bit more specific therapies not later some other important sets

so I expect you to remember that the set bold-faced end this will always mean exactly the same thing in this course it’s the set up in this textbook can’t be both effects this is how and it’s described there are two other sources which include the number zero in the natural numbers we won’t write so it’s only positive numbers but it could be non-negative integers so some people will start at 0 1 G ba ba but any text any text that you do some of this will actually tell them what you want they need your textbook will say is 14 so sometimes include 0 the natural numbers sometimes they don’t this is called the natural numbers natural because they were part of the first things that naturally come as far as numbers because we want to another set is called Z which is the integer right so this is like zero so that’s the natural numbers the negative natural numbers and zero because the integers everyone who finds that extremely there’s also a set called Q full today’s Q this is it’s very hard to actually write this any elements in a predictable pattern so we normally Express cute in this one set orientation basically we defined it as all LMS that by a over B such that a and B are integers but B is non zero so this is called rational numbers or is they’re often affectionately call fractions so the set of all fractions is a thing we call it the set key actually don’t know we started while called keep what we call the two in this book we use I use a notation our base are two real numbers the specific description of the real number is also pretty difficult to describe but we usually think of it as a number line or all possible decimal expansions there are tons of esoteric ways to describe it real numbers the completion of Q is one way but you know that so we won’t get into anything terribly complicated as far as that’s concerned the set I is the set of irrational numbers cannot be rational what progress comes from the word racking up some ratio you’re taking a ratio of integers you’re asking on this means they’re not the ratio these are the real numbers examples the spirt of 2 is not rational that is an element I knew in fact true this at some point in this course

hi is it an element of I number these are all numbers that we cannot actually express them as a fraction it’s actually possible to write it down as a ratio of integers and we’re actually improving that at some point they’re not many for complex numbers too much but I might as well tell you what they are so see this is the set of all numbers like this that a and B are real numbers and I squared is minus negative 1 so the number whose square root is minus 1 these are called complex numbers we invented up so that all polynomials are through in real numbers we’ll actually have a solution there are things like our plus RR plus 0 so in fact positive real numbers positive ones including zero this one patient pretty typical like I think I can do it with Z for example which is just a fancy way of saying it so I can almost look like a plus I could put up – Danine I owe a lot the negatives or something so these are standard letters to represent these sets at any point if you see me write this or this or this or this or this this is what baby always need this I will never use this symbol to represent any other kind of set other than these lists and you need to know what these definitions are so it will be a tea leaf accent for definitions on Eclipse like I’ll save what is this define cute there is a notion of spice we reuse this cardinality seed or card knowledge arrow that means the size of this set right describes by what we call partners the set is called why not if it got me actually pretty good notation into that this actually have some values but usually don’t confuse it outside vice a Java MapReduce letter notation so you see a second one right sardonic loss say and he said for you to describe the

number of elements in it by some integer we call that a finite set otherwise the step is called infinite so internet basically means not fine any rules one two three then I can write that the size of P is just exercising p4 right look at all the guys that are separated by commas Sallu see one two three four five yes so it C contains a set that contains two elements but as far as item C is concerned disguises well this is called it’s an infinite set but the thing is this is sort of a big boy level math classes that would have been fine but for us now this is not made because as we will see when we get to chuck or ten on this textbook there are different sizes to infinity not all of it needs not all the infinity is part of the same so writing about infinity is actually too big no guess what there’s a Hebrew letter that’s used to represent this call Allah tell it not is the size of to be natural numbers you look at the size of real numbers then it’s you need to know it see the size that sees roughly the size of a tune to that amount and it is actually bigger than the size of an apple so all of these sets are infinite but actually more real numbers in substance than your passion yummy this will this is called apartment number 4 the natural numbers and see is the cardinal number for the real number let’s call it letters because we can’t be describe it specifically we just know they’re both infinities but that infinities bigger than this one with the empty set 0 there are no elements in the empty set what is the size of the set containing some have mentioned other currently with all these definitions

there they’re just we’re going to be using all these phrases in everyday language from at this point on and I just want to make sure everyone’s on the same page and everyone will know what I’m talking about when I say cardinality subset I don’t want that to be baby so we’re going through step by step we say if every element is also a logical notation this would mean that if X is in a and applies X is in B top office later respectively tell you what that is and that’s chocolatey so that’s what the subset is so for example and we can say a is a subset people notice that a is not a subset of C why is it not a subset of C because one is not in C right so there’s an element in a that is not contained in C so in this case yes right we often say nobody’s allowed that it’s sort of makes that intuitive it is contained so yes you can literally find everyone 1/8 is also a member of a date is possible that is actually equal to B which top I will actually measure what it needs to be equal that’s what slope bar that CDs there’s possible to the subset would not be equal of course by example but he’s not equal to meet me they’re not the same set just to see alright so I write the bar on 31 I know quality’s a possibility or it’s even known and I can write this when I know for example that is not intercept we call in a proper subset in this case something that’s very important notes

for any set the empty says our subset of Lee and the set itself so you can also think of a subset as being a set where you actually choose some of the elements to put into it right there might have a set B and I create another set right in there thinking elements from B and putting into this set I can call that new set that I’m creating a subset it’s possible that I choose no elements in which case I get the empty set so that’s not possible subset it’s also possible that I Tunes all the elements from a set so the entire state is also associated again we can prove this it’s not hard so online I hope you don’t have proof yet so let’s just just know that where are the subsets of the empty set okay so the empty set is actually a subset of the empty set because of each of those conditions but of course in every set is not be careful things I thought that attention to detail very important Oh there’s a notion of a universal set so for each different polymer each different context who have a different universal set this is basically where all possible members that you consider deliver so for example people’s numbers then I can think of the natural numbers as a subset of you think of the real numbers as a subset or I can think of you as a subset of you for this particular it’s the set of elements right if I said you was a set of letters and these are not sources but you never want to it turns out that sets can’t get too big well they can get really big infinity but not too deep like for example there’s no such thing as a set of all sets there’s a lot of paradoxes we’re not gonna get into that we’re always been you know we’re creating set by peeking from some larger set that is always not chuckling we can get pretty big

we’ll get back to this when I talk about in diagrams what else do I want to say I also wanted to mention something about intervals so we have something called open door this is something like maybe you’ve seen this before stepping into all the nominal this is basically equal to the set of all X such that X is between a and B and X is a real number or this is called open interval there’s also something called the closed in which we put this notation just a set of all X such that it’s less than or equal to there’s something called a half so you cannot have the open on the left or to be open on the right set of all X this is less than navel this is the set of all X such that X is less than equal to this is now finds the intervals equality these are equal we use just equal sign to distance relationship it means and this is gonna be important

there that a is a subset of T and at the same time B is a subset of people if you have all these conditions people know about the same time it turns out they have exactly the same elements sometimes we need to prove their quality and that’s how we’re gonna do it we just proved that including those both ways the idea of power call it PLA so it’s a fancy to you I can’t write this fancy no all right so it’s called a team of eight and so this is just the power set of a set date examples we’re not going to talk about here more

than happy interesting stuff approaches stop here after adidas examples and they’ve got to put this next time which is on Wednesday the first one what is behavior right so notice that P of a is itself a cent right so it’s the set of subsets it’s called healing so the empty set is a subset of P what else is a subset of itself is there anything else those are all possible subsets so if I actually what is the size of a what is the size of the parcel what about here what’s the possibility here the set secondhand 870 feet and the mean itself besides me the size of the powerset about here is the empty set what’s up our scent the set containing any set is between set okay so what’s the size of eight zero was the size of the power said about this one power set of a movie so we did the first and the second so this is empty set and descent we can do the second and the third element so this will be the set containing the second thing in to set and contain the empty set and the second – the empty set in a set and we can also do set the pin all three elements right the set itself very there is second element of what is there

anything else right we also have to do the assessment in just one of the elements right so that would probably be the first the first list part of the list so we have that anything else so we have a subset that contains none of the elements we have the subsets that contain one of the elements each but the subsets that contain 200 calories each and whoever subsets that contains all three elements is there anything else I should get what’s the size of Game three what’s the sign of the power one two three four five six seven eight there are eight elements okay so make a guess if you know the size of a set is a finite set of size n what’s the size which is the size of these stupid otherwise what’s your conductor a conjecture by the way is just a fancy forgets educated guess what would you guess the size of P of a is due to the end there’s probably a nice conjecture here we have a 1 unless T to the 1 is to characteristics or 2 or here at 0 elements teams of the 0 is 1 here at 3 elements 2 to the 3 is actually 8 so you actually make a guess but what if this breaks down at some point what if it’s not always two to the end how do you justify that your injector actually made sense give you an arm any other there’s a there’s a more straight for sophisticated argument that we will be able to do when we get to chapter six right where we do induction that would be a really nice straightforward way to prove this but there’s actually another argument here how would we think how we try to convince me that tea to the enemy the side of the powerset turns out that doesn’t work so convincing it turns out me convincing someone of a general statement by from a finite number exam of the examples is actually kind of tricky because then you can be like you know what one is crying three is prime by this Prime all others are prank right entry’s no it’s not true so exactly you’re not good enough to convince me of this bicycle works here right give me an argument yeah how do you have two sets yeah but I didn’t repeat the null set for every sector think of how we constructed the subsets there was a set of 1 and 0 elements the set that contains 1 the set that contains 2 there the 7 contains 3 so pretty much to make a subset how to actually do it I pick elements on the set and I put it in another set I can pick no one in which case the subset is the empty set or if I think only one of the elements and clear this set I get these guys or I can pick two and these guys sparking all three and I get this stuff so it turns out to create a subset what am I actually doing I’m choosing

elements right now for each element in the set how many choices do I have only two choices I can decide I’m going to pick this one I’m not going to take this one drink so the justification might be something like to create a subset choose elements from 8 for each element the two choices which is it or not there and we have tuned to the end choices right so if you have one two three four all the N elements but here you have two choices or two choices for each element then there’s something about the multiplication rule which you probably see you just multiply all these elements 2 times 3 times 2 times argument to end an argument we knew this symbol the more archaic ways an argument is that you write the letters it’s actually a lot for quarter after in Streatham which means as was to be demonstrated this is 2n an argument so you see this is sort of what this class is about most of all be able to convince someone or something like in general not just like oh we know it works in these three pieces so at all those words that’s actually not good in it because that breaks down there are many examples where it works some of the time but not all the time and we’d like to know when something works all of the time so a lot of times you’re going to make it guess at something and you’re going to justify and so that’s mostly what responsibility above of course they’re going to be other but that’s sort of the idea that’s where we’re going and this is next under questioning

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