Lecture 1: Sets, Set Operations and Mathematical Induction

[@faizanmansoori]

I am crying right now. I am an economics student and I have been waiting for MIT OCW to release lectures on Real Analysis for so long. My whole mathematical economics training has been due to MIT OCW from calculus, linear algebra, differential equations to statistics and ML. Thank you MIT.

[@MrScattterbrain]

I'm in the same club! Have bought myself a beer to celebrate this release. Hope MIT will publish Measure Theory some time soon.

[@DanielSonghere1454]

They got a good series from Harvey Mudd. But most courses vary on topics covered

[@saaqibmahmood5495]

@@DanielSonghere1454 The Harvey Mudd course leaves out integration and what follows thereafter.

[@jaredvv86]

@@saaqibmahmood5495 also the video quality of those Harvey Mudd video is less than ideal

[@jaredvv86]

Mathematical econ is so much fun right? That was my major in undergrad.

[@vhiremath4]

Wow this guy is an incredible teacher. Constantly states why things are important. Has a sense of humor. Builds up logic so that things flow vs. a spattering of random facts. 💯

[@victorrodriguez5981]

after years and years of no well known university uploads a full course in this topic, it had to be MIT Thanks Again MIT from Bolivia!

[@Deliriummary]

This is incredible!! Thank you so much! You can’t even imagine how much I appreciate this. I would really love to see more pure math content like topology and abstract algebra!!

[@darioinsi9370]

Maths is always badly taught

[@mississippijohnfahey7175]

@@darioinsi9370 always?

[@darioinsi9370]

If it is not always, it is almost always or both. Altough this at glance can sound like a triviality or simply an exaggeration, it really is a fact. If you allow I can give you a quiz and after we will have a good discussion on how math is badly taught. The quiz is that: To ask at least 10 people who have studied maths at a time of their life whether in primary school, secondary school or University to explain you at least vaguely WHAT IS A CIRCLE. If you find 4 people who are capable of defining a circle at least vaguely by using words like center, point, radius, equidistant ...at that time I will say that Math is well taught. Another question you can randomly ask is that if 9.834 millions km square is area of USA, what does concretely mean?

[@mississippijohnfahey7175]

@@darioinsi9370 I think the average math teacher does not give students a reason to care about the topics. Just calculating something because you're told to does nothing for most people, especially when they know a program can calculate it for them. Students need an important and difficult problem to work on that forces them to learn in order to solve it. 101 problems at the end of each chapter that are essentially the same problem with different numbers are intriguing to almost no one. Unfortunately, the kind of math that is taught this way comes after calculus, which is usually the gate that keeps most people out of higher math. However, I do believe there are good math teachers out there. I particularly enjoy MIT's courses, because when you combine them with their physics courses, you wiggly understand why it's significant. However, I do think MIT professors are too find of doing all their work by hand on the board. It's great they can do it, but showing how to make a computer do it would be much more beneficial, since that's how pros do it (in my industry, at least).

[@robbertcox6170]

Thanks for the early Christmas present, OCW. I'm sure this one was on the wishlist of many 🙌

[@lezeuvete2508]

After all theses years, MIT finally decided to post pure math videos. I haven't thought such a day would happen...

[@vinylflouring]

This guy has the style to teach this subject matter effectively. Best analysis content on KZbin

[@dtung2008]

Prove induction theorem by well-ordering property! Nice and clear.

[@digitalfroot]

great video. A discussion woth my dad about him failing real analysis is what played the seed of being a math major. We had that conversation probably 6-7 years ago and now i’m a junior in college pursuing a math degree. So happy to have this information at my fingertips. I get to watch these lectures in my own time and learn a lot by myself

[@preciouspetermasula6315]

being a maths major student, and lucky enough this course we are taking this semester, this is really helping me Dr.

[@manoranjansahu7161]

I have been waiting for this for 10 years!!!

[@maxpercer7119]

I love this mathematician's logical way of thinking.

[@tuongnguyen9391]

I am an engineering student and I have been looking for this for such a long time. Real analysis, functional analysis. I hope in the future there will be complex analysis. This is extremely useful for digital signal processing stuff. One strange thing is that currently all the machine learning stuffs are not design for things that have complex number inside them

[@CSWG-d9e]

Yes, could you please upload a lecture series on Complex Analysis, @mitocw?

[@mississippijohnfahey7175]

Hey MIT, thank you so much for posting this! If you want to improve the sound, try using an expander (a compressor set to a less-than-one ratio). To make the chalk more visible, the contrast could maybe be turned up. Come to think of it, the expander is just like audio contrast! After expanding, you may want to apply a maximizing limiter. The Waves L2 plug-in works great ($30), and you can do all this audio work in Ableton Live ($250 for students). My mentor at work told me a few months ago I need to learn analysis. Since then, I went through 18.01, 02, 03, and 06 to refresh, and to see this lecture series up now is just perfect--thank you again!!

[@nbooth]

Since he's doing it without a live audience anyway, I wish he would do it with a lightboard. It is much easier to read, doesn't smudge when you erase, and it just looks so freaking cool, like the equations are floating in air! But this guy is an amazing teacher so I'm not complaining.

[@vazn4143]

Thank you so much MIT for providing high quality courses like that OMG ! And i'm entering my math major next month, that's a GREAT TIMING😄

[@carlosmateosamudiolezcano2463]

Finally a real analysis video lecture series from the OCW :'). Thank you

[@saaqibmahmood5495]

O MIT OCW, please release videos onto KZbin of the 18.100B, 18.100C, 18.700, 18.701, 18.702, and 18.901 courses!!! And, also of the complex analysis courses taught at the MIT! Please do seriously consider this request.

[@turokg1578]

a day after i watched this lecture. the proff in my uni proved induction too. and i can say that this guy ROCKS.

[@every1readthis]

I love it. I get to go MIT for free learning.

[@yan-amar]

Don't feel too akward professor, there are people actually listening on the other side, maybe even more eagerly than some of the students that get to be there in person.

[@user09832]

Yes

[@squarerootofpi]

Well this course was taught in 2020 so the real MIT students are all probably listening from home like us. It was awkward for almost everyone back then.

[@H3XED_OwO]

agreed, he is a good teacher and i like math

[@diegolainfiesta]

I got dizzy by following the video, the cameraman is constantly following the teacher, instead of leaving the dashboard view. Because here are no students, he is writing pretty fast without any interruption. It is so say, I´ve been waiting this course. If you have the chance to fix it, it would be great.

[@zukamakanesada1267]

I really appreciate this course and I hope complex analysis lectures will come out soon

[@cswanson4476]

47:53 - 48:36 Chilling. Doesn’t want us to suppose the rules of logic are as obvious as they seem. And since he’s been quite sharp up til now, it kind of startles you to hear him warn of something “hairy” that stalks us. Kind of spooky, like we’re about to listen to a campfire tale. It’s almost cinematic in a way, as uncertainty about the rules of logic creeps into his monologue, he wanders into a thicket of upturned chair legs.

[@every1readthis]

Remember to increase playback speed to 1.5x’s.

[@ahsanaldaw2292]

I wanted to do that then I realized it was already on 1.75

[@EzraSchroeder]

You can even do 2x as long as you are really processing it because Dr Casey's enunciation is perfect

[@j.adrianriosa.4163]

@MIT OpenCourseWare, and Dr. Casey Rodriguez: thanks a lot for this. Now that it has been released, more and more people over the years and over the world will continue to benefit from this.

[@anodynemathematician4194]

Glad to see this posted. Hoping of going to the Eastern Half of Australia next semester so that I can study this.

[@martinhawrylkiewicz2025]

Firm foundation in logic, set theory, and proof writing from books like Book of Proof or How to Prove it will open up a new vista of different possibilities

[@abdurrahmanlabib916]

Waiting eagerly for 18.100B !!

[@bacobjeatty2777]

Just read some Rudin on the side and maybe go a little deeper into some basic topology

[@patrickthibault8119]

He did a great job given no students were in the room.

[@Rerxxy]

Thank you sir, sending love to you from Nigeria 🇳🇬

[@sysyphenf8ewtfr603]

Everyone is excited for this course but only very few will complete it from start to finish.

[@mississippijohnfahey7175]

And many less will understand it. But that's okay. If you don't finish, then you probably didn't have a good reason to do so, and it often takes many times and many ways of seeing something to truly grasp it

[@husam1919]

Some people will take the whole thing, some people will use the lectures as a replacement for their profs, some will use them as a refresher, there's no pressure let the people just learn what they want and at whatever pace that suits them.

[@هذاأنا-ذ3ث]

This is the case with just any educational lecture series. Usually 10% finish it beginning to end and about 1% would watch more than once.

[@AbhiKumar-og2jv]

Please upload the lectures of 18.100B Analysis I it is very important concept and we need it.

[@gloriosatierra]

Actually in Mathematics that box means completion of a perfect square: like saying all things are squared now between me and God. Tip: spray water at the eraser, erase, and dry clean with dry towel the board for vibrant calcium carbonate and no more dust. 🌿

[@davidbellamy1388]

Thank you so much for putting analysis online!

[@xavier.antony]

Why the camera is wobbling? Why not focusing on the board? more distractions

[@brucelu4782]

Thanks for posting this course, the instructor is great. If I may, there is only one request, in the future if the camera could move less frequently, the camera is following the instructor too closely, making me a bit dizzy.

[@shashanks.k855]

I have been waiting for this since the day I was born ! I shall achieve the Buddhahood once OCW releases the "Functional analysis" series.

[@shashanks.k855]

Nirvana time ! kzbin.info/aero/PLUl4u3cNGP63micsJp_--fRAjZXPrQzW_

[@jacksonmadison9994]

A complementary Real Analysis recitation series would be much appreciated.

[@cswanson4476]

4:13 Can we define a “collection” of elements as an enumeration of those elements (i.e., a “roll call”, either explicitly and individually, or by describing necessary and sufficient properties)?

[@Y2B123]

Do you have any plans to release 18.100B lecture videos? Thanks for what you have been doing for higher education!

[@mitocw]

Currently there are no plans to record videos for 18.100B. There are non-video course materials available for 18.100B that you are welcome to review: ocw.mit.edu/courses/18-100b-analysis-i-fall-2010/. Best wishes on your studies!

[@kingarth0r]

perfect freshman course; amazing

[@Random-sm5gi]

Yes Dr Rodriguez, thousands of people will listen to the your lecture. Thank you for the lectures! Must be more awkward if you imagine yourself lecturing to number of people that is of stadium size.

[@nagarajuchukkala9538]

The camera movements are making me dizzy. What is the need to move the camera when the professor is just trying to move out of the way of the board? Shouldn't the board be the focus of the camera?

[@nicholashowland3809]

Solid material, but what is happening with the camera pans?

[@funfactory463]

@mit the course is very very helpful for understanding course. But 1 request please don't rotate camera like this. It's really irritating. If professor remain some second in corner of frame then it's ok but camera rotating for this is irritating. Just keep focus on board. Plz

[@jeffaugust8249]

Usually the courses in real analyses miss the whole set of numbers between the rational set Q and the real set R, the set of surds which themselves form a field, they are the elements containing algebraic combinations of roots and nested roots. They are extremely useful. I told my real analysis instructor about this and he dismissed it.

[@rhodesd]

For anyone wondering about the last claim, how $1 + (m + 1)c$ could be the (n = m+1) case, plug (m+1) into the inductive assumption, which results in $(1+c)^(m+1) \ge 1 + (m + 1)c$

[@riverland0072]

going to have to take this whole series cos Real analysis was probably the hardest course I took at UNN

[@Nikhillbt]

I'm listening.

[@neverbendorbreak]

I love this guy!

[@KingZero69]

thank you for releasing this 🙏

[@kj8002]

at 31:40 why it's written x doesn't belong to B "AND" x doesn't belong to C? Because in the definition of B union C, x belongs to B "OR" x belongs to C right? So, if I follow the (B union C) definition, I think I won't be able to complete the proof?

[@rohielahmad3945]

Think of the Venn diagram. If x doesn't belong to "B" Union "C" then x doesn't belong to B\C, C\B and B Intersect C.

[@thatguyrandy1859]

35:21 The box does mean something, it means to release the happy chemicals in your brain.

[@danielbautista7]

Could have used this decades ago but better late than never. Can we have a lecture series on Sets for Mathematics too

[@studentsfactory530]

There is a correction in the set definition as you missing the word well-defined. All collections of objects won't become set if it is not well defined. Example name of best 10 players in a country is a collection but not a set

[@alqadaffi]

No news to worry. We are listening.

[@hsenagrahdeers]

Isn't the definition of odd numbers at 14:35 a bit restrictive? If they are to be defined as all 2m-1 where m is a natural number, aren't we limiting ourselves to only positive odd numbers? I believe it's better to define them as all 2m-1 where m is an integer, am I missing something here?

[@confidosine]

you're absolutely correct

[@yangyang3162]

Thank you for the great lecture but needs a better cameraman. Cant see the board cuz it is cut off and camera follows instructor around every moment you cant see the board. Shows too much of chairs and too little of the board

[@jasonr702]

Small point - in the last proof, the theorem is for c >= -1, but we only proved for c >= 1. It's trivial to prove, but still worth noting, I think.

[@aashsyed1277]

No we proved it fair all c not 1

[@dtung2008]

Contrary, the proof doesn't need the condition of c>=-1, it is valid for all c. A little bit not precise.

[@rohielahmad3945]

Awesome work MIT OCW. I have been looking for a quality course on analysis for a long time. Even though professor did say to not get hung up on the induction theorem proof, I am hung up on the proof. One statement in particular - Since P(1) is true how did that imply that 1 is not an element of set S? Is that from the statement in the beginning that S = {n: P(n) is not true}?

[@austinkim9402]

Why wasn't this here 3 years ago when I was suffering

[@أحمدالدسوقي-ت9س]

Now you have the advantage of getting the most out of it since you're the one who suffered.

[@frederickaoay5097]

Kindly please make the camera stable also should apply close up, can't see clearly.. I'm getting dizzy... Plssss🙏🙏🙏🙏

[@tzachs_]

Well done .....MIT!

[@abhinandps3034]

Spectacularly potriated

[@MiguelOliveira-ec2ie]

Nice start ,

[@jonathantolan3503]

The definition of odd numbers doesn't include negative odd numbers

[@casper6349]

21:56 "union" instead of "intersect"

[@SnackFatson]

are all of the mistakes top-down finger-swipe dual-purposed logic!? Thank you for letting me see that!

[@ryanchiang9587]

bolzano-weietsrass theorem

[@jmcdonne]

Interesting definition of odd numbers at 14:30 as being only positive.

[@georgeyang3228]

1:13:56 I am not sure why c has to be greater or equal to -1.

[@MrScattterbrain]

This is required to preserve the sign of inequality. Check 1:10:03.

[@georgeyang3228]

@@MrScattterbrain Thanks. That helps.

[@minglee5164]

Such a great course

[@electrophon]

Lovely lecture.

[@mkkkk1643]

Little bit disappointed, thought it would the course that follows rudin's book(18.100C),but still appreciate your upload MIT!

[@aniruddhnls]

What is the reference book for this course? I couldn’t find this information on the course page.

[@mitocw]

From the syllabus, "Lebl, Jiří. Basic Analysis I: Introduction to Real Analysis, Volume 1. CreateSpace Independent Publishing Platform, 2018. ISBN: 9781718862401. This book is available as a free PDF download. You can purchase a paper copy by following a link at the same site."" ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/pages/syllabus/ Best wishes on your studies!

[@Tradingkiduniya290]

Good informative knowledge 🙏🙏🙏

[@siddeshkhedekar2286]

Thanks prof.

[@thevikingwarrior]

I was trying to understand these logic symbols, for personal use. I wasn't taught them at school etc, and I came across the fact that many symbols are used to mean a number of different things; and so there seems to be no universal system for the symbols meaning. That makes me feel super-annoyed. It can't just be simple and obvious can it? 😵‍💫

[@muhammadnawab3959]

F set is the collection of objects then how can u say that empty set is a set

[@napluto2031]

14:44 aren’t the odd numbers also negative

[@allandogreat]

a new course, we are your students.

[@kilgoretrout6810]

He's left-handed, so his body doesn't get in the way of what he's just written.

[@unnamedchannelowouwu]

Can someone tell me what's the level of this course? Is it good for pre calculus? I just finished high school and I don't wanna fail in college (I'm going to CS) will this work for me?

[@Bryang229]

Hey I would say try to avoid real analysis and focus on the fundamentals of calculus for now, 3Blue1Brown has a great series going over the intuition needed for calculus. Personally I’ve just finished calc 3 in college and this high level math still boggles my mind so don’t over confuse yourself just yet :)

[@الرياضياتالعظيمة-ز6ك]

First course of analysis or algebra of the first year in University

[@paigedote954]

great question! real analysis assumes some familiarity with calculus. while you could technically learn calculus from real analysis, it is much nicer to digest if you have some experience with that

[@阿波羅-y9f]

This is usually a post-calculus class, so I'll suggest you to come back here after you are familiar with your calculus. As for pre-calculus, my suggestion is to go directly to the pre-calculus section on Khan Academy and make sure you are familiar with all the stuff there. Then you can take the 18.01SC class (available on this channel) to learn (single variable) calculus.

[@rockinray6197]

this fellow doesn't engage very well. try the chinese math teacher Eddie Woo on KZbin. he discusses a few puzzling ideas on your level. there is no proper 'level' in math.. its more of an art.. like arte' its meant to be revealing, and MIT is 'research level', a step removed from educational study. thx for your question

[@nikhilchaudharybirbhan377]

Love professor Casey. Can we get lectures of linear algebra, matrix algebra, complex analysis also?

[@mitocw]

A good way to find our materials is to go to ocw.mit.edu, do a search, and filter the results e.g. ocw.mit.edu/search/?q=linear+algeba. Best wishes on your studies!

[@nikhilchaudharybirbhan377]

@@mitocw No latest lectures on it. During covid classes, a lot more classes were likely recorded. Would be great if you can upload video lectures of lot more math courses.

[@qbtc]

Will the rest of this course be posted also?

[@paigedote954]

the rest of the course can be found here: ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/

[@anonym498]

Should i already have experience with proof writing?

[@jerryhou9402]

It will be great if the answer to the psets can be posted same as exams as for studying the way to write proofs!

[@natty_ankrah]

22:01 Union not intersection

[@mateocamargo7437]

Can someone explain the inequality from 1:12:53 to the end? Thank you

[@vansf3433]

Discrete mathematics can give an insight to human -invented notions of mathematics, but cannot escape from the same common problem of human subjective interpretations in human -created notions of mathematics either

[@lebesgue-integral]

Can I make a "complement" to your video? That was great :)

[@Sen7channel]

Wah real analysis----!!!!

[@henrikhansen9358]

Keep up the good work🙏

[@oxygen2671]

I don’t understand why is S=an empty set .in the proof of the principle of mathematical induction can any one tell me why ?

[@rajvardhansinghsisodiya1095]

which book are you using

[@mitocw]

Lebl, Jiří. Basic Analysis I: Introduction to Real Analysis, Volume 1. CreateSpace Independent Publishing Platform, 2018. ISBN: 9781718862401. This book is available as a free PDF download, see ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/ for more info and materials. Best wishes on your studies!

[@withoutpassid]

Was this lecture recorded during the Covid social distance period? I don’t see any students in the room.

[@grace16.02]

Yes, I believe so. Description says fall 2020.

[@UltraMaXAtAXX]

That whole first few seconds was me in Fall 2020 as an instructor. :(

[@siddhantmisal4115]

Finally

[@casper6349]

Which textbook does the course use as supplement to the lectures?

[@leonardocaetano6307]

What is the text book of this course?

[@mitocw]

The textbook is: Lebl, Jiří. Basic Analysis I: Introduction to Real Analysis, Volume 1. CreateSpace Independent Publishing Platform, 2018. ISBN: 9781718862401. See ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/ for more info and materials. Best wishes on your studies!

[@sportmaster2586]

Is this first year ?