Bro yes. I’m in college doing a business course, which in my opinion is pretty easy. It’s common sense and taking notes. Three people dropped out of my class last year alone. We had a class of twelve.
Something I’ve notice is that classes considered “hard” usually don’t tell you the background knowledge you should have before taking the class. I find this really prevalent within chemistry since nobody tells you how much a good grasp of physics concepts underly chemistry
Yeah because they want cash so there not going to say fck all man. Thats what happens when you let universities have total control with no locks on them
Honestly you don't really need to be fully knowledgeable in physics to understand chemistry. HOWEVER, knowing and truly understanding basic physics concepts and being able to apply them in various means will undoubtedly make chemistry easier to learn, understand, and retain.
I've looked at the homework and the curriculum for the class and honestly it's not an incredibly difficult class. It's difficult mostly because the students in the class have to completely deconstruct everything they know about mathematics which is a lot harder than straight up just learning maths. If you want to you could actually create a jerry rigged version of this class on your own lol. For the first semester of the class go through the entirety of a linear algebra and abstract algebra textbook of your choosing (for abstract algebra I've heard great things about Judson's textbook and I really like it) then for the second semester go through all of Michael Spivak's Calculus textbook and congratulations, you've completed math 55. If you want to make up a lecture for the class then look up "real analysis", there's a playlist by "brightside mathematics" who does a great job at explaining analysis. Watch videos 1-14 of that series and you'll get what's a single lecture in that class for the analysis portion of it I think
Yeah exactly. 45 out of the 60 in my microbiology class dropped out less than halfway through. Having a lot of biomedicine under my belt now I can safely say that any relatively hard class will have a dropout rate of atleast 50%
There are classes at every university made intentionally to weed people out. You want the best of the best in certain programs so having those courses are essential. 50% drop out rate is pretty normal and good often times If most people could do any class they enter, then degrees would be meaningless lmao
@@MasterMenahem you don't understand for classes like those you require a test before being allowed in to make sure u have the skillet necessary to learn the info a high failure rate means the teacher is shit and the ones who passed didn't need the teacher to learn this info they could have done so with similar resources and no teacher
1. To prove that the discrete metric is indeed a metric, we need to show that it satisfies the three properties: non-negativity, identity of indiscernibles, and triangle inequality. a) Non-negativity: For any points p and g in X, d(p, g) is either 0 or 1. Since distances are non-negative, this property is satisfied. b) Identity of indiscernibles: d(p, g) = 0 if and only if p = g. This property holds since the discrete metric assigns a distance of 0 only when two points are equal. c) Triangle inequality: Let p, g, and h be points in X. Since d(p, g) and d(g, h) are either 0 or 1, the triangle inequality holds. With the discrete metric, subsets of X are open if and only if they contain each of their points as isolated points. Every subset of X is open since each point in a subset is isolated. A subset of X is closed if and only if its complement is open. Since every subset is open in X, the only closed subsets of X are the empty set and X itself. Every subset of X is dense since every point in X is isolated, meaning there are no points in X that are not in any subset. 2. To prove that do is also a metric on X, we need to show that it satisfies the three properties: non-negativity, identity of indiscernibles, and triangle inequality. a) Non-negativity: For any points x and y in X, d₀(x, y) is defined as d(x, y) / (1 + d(x, y)). Since d(x, y) is non-negative, and the denominator is positive, d₀(x, y) is also non-negative. b) Identity of indiscernibles: d₀(x, y) = 0 if and only if d(x, y) = 0, which happens if and only if x = y. c) Triangle inequality: Let x, y, and z be points in X. We have: d₀(x, y) = d(x, y) / (1 + d(x, y)) d₀(x, z) = d(x, z) / (1 + d(x, z)) d₀(y, z) = d(y, z) / (1 + d(y, z)) Using the triangle inequality for d, we can show that: d(x, y) ≤ d(x, z) + d(z, y) Dividing both sides by (1 + d(x, y))(1 + d(x, z))(1 + d(z, y)), we get: d₀(x, y) ≤ d₀(x, z) + d₀(z, y) Therefore, the triangle inequality holds for d₀. A subset of X is open under the metric d if and only if it contains an open ball around each of its points. Since the metric d₀ is defined in terms of d, the same subsets that are open under d are also open under d₀. The metric spaces (X, d) and (X, d₀) may not be the same as metric spaces, but they have the same open sets. The metric space (X, d₀) is always bounded since d₀(x, y) is bounded by 1 for any x, y ∈ X, whereas (X, d) may not be bounded. 3. Let's examine the given options for defining a metric on R: i) d₂(x, y) := √(x - y): This defines the Euclidean metric on R, which satisfies all the properties of a metric. ii Just call me the thinker 😗
@@bigchocolate1118 no shit but when you make it to the point where you don’t teach how you got there then you ain’t teaching you just putting out the assignment. You don’t learn that way. I had professors who would just put out the work and the assignments but never really taught. Like how am I suppose to learn that and do the assignment when you didn’t teach us shit smh Had computer science classes that were at the intro level c# class and the professor would never show how he got to the end point. Like mf how did you get there? Like this is an into class and I really need to know that! Please show me got damnnn it 😂
@@jmemixx exactly. At that point I'll just drop out and teach myself since the people who are paid ass loads of money to sit and do jack shit and expect us to use sources to learn won't do it. Literally had professor's like that and it pissed me off to no end especially since they're stem courses.
@@bigchocolate1118 That's not the point if they were actually taught to the point of understanding then it wouldn't be half dropping out. they're Harvard students, they are smart- unless half of the class didn't study at all or the least smart in Harvard then the number of dropouts shouldn't be that much unless they're not taught well. Yes, like you said they're Harvard students; do you expect them to be dumb and not study at all? Lol, you think half of the class are 'shit students'? Well, I don't think Harvard would accept that many 'shit students'?
In my opinion, having a class like this is ridiculous and not a flex. I’m not sure how it is in America, but I’m Australia each unit should expect about 10 hours (including contact hours) of work weekly because 4 units makes a full time study load. That unit better be worth 4 times the credit for it to be worth taking.
@@Starlight_is_cool math isnt more hard tho. Fine arts are. The only reason they are easy to pass is because of what also makes them hard. Like languages, they arent logical or structured at all. But that also allows you to pass things like art. Math is super easy because its 100% logical so it makes total sense. Nothing just “is” without a reason(or at least one that is a bit more advanced than your level)
@@o_sch Okay but consider that it maybe easy for someone like you because you need logic and structure, whereas for others, fine arts is easier because it doesn't have a definite answers for the most part.
Ooh metric topology was one of my favourites in undergrad! Those first two questions are pretty solid ones too, not too hard but definitely need a good understanding.
Well, this just gives me flashbacks to today and everyday for the next 2 years... cause a lil brat thought "oh, I like maths, so why don't I pick my nations hardest most intense maths program even though I hate homework"🥰
Math gets really fucking hard. If you want something relevant to you, please look at how rendering engines calculate light paths, and enjoy the math rabbit hole.
Tbh Ive done those concepts and I passed each course that taught each topic with a really good grade but I can see how ppl might find it hard, it’s a lot, and also a lot of problems. I’ve had a similar class that the professor gave 1 assignment each week that was 15 double sided pages it was for a discrete math course it was so much work that I ended up dropping another class so I can put more time into that class
honestly, most math grad can do those questions. The course is challenging because it’s an intro course that compacts ~5 course topics. However, it’s not difficult overall
Just looking at this assignment gives me PTSD... It may not be particularly difficult, but going into a maths course expecting highschool 'algebra', you will be sorely mistaken.
I am the type of person who will take course just to see how hard it is and since I hate dropping courses, will stick with it. And well probably fail it since I am an idiot who took it even when others warned me.
Don't take school to heart take take those grades to heart this school system wants you to hate yourself the whole reason school was made is to train you to work like a slave they want you to work for other people instead of working for yourself the guy who created the school system said he wants a world of workers not thinkers they want us to physically do hard labor work for money even tho we get paid nothing school is useless unless your working towards a job that requires school then do that but if your job requires no school then you don't need school most of the things school teaches you is random bs that we will never use in the future 90% of the time like history they teach us meaning less things to just distract us the teachers even know it they just do it for money and if you wanna leave school or try to drop out you will get involved with the court and you or your parents could get jail time even tho you don't need a diploma if you know you got the mentality to do your dreams do it just don't doubt yourself
1 = The discrete metric is a metric. All subsets of X are open, all subsets of X are closed, and all subsets of X are dense 2 = The metric space (X,d0) is always bounded 3 = The metrics that define a metric on R are ii) d2 (x,y, and v) d5 (x,y) 4 = The given statements i), ii), and iii) are proven. Two metric spaces (X,d) and (X, d0) are isometric when there exists a bijective isometry between them
When I was a kid I decided to major in math because geometry and computation were fun. Only thing I was good at. By the time I hit Mathematical Analysis in college (a math class where your TYPED homework answers with symbols looks like essays 🙂) it was too late to switch majors. That’s where villains are made, but I still earned my degree all the same~* if you’re a student mathematician at ANY level, keep pushing!!!!! Much like anywhere else in life, only the persistent last.
@@guccifer764 indeed in my case we were 50 students and only 17 finished the course, the class average was 57 , biochemistry 2 analytical spectroscopy and gaz based ccm
Because it can only really spit back information, not actually process it in a conceptualized way, yet. That would require an Ai with self learning and recognition. That is a few levels above chatGPT or whatever u are using. 😅
@@cielonehellofaservicedog4648still though, the fact a Ai that has access to a lot of human knowledge, yet still can’t figure this out is funny to me at least.
These problems really aren't that difficult. They are just university-level math. Not easy, but not impossible either. I'm not good at english math, but I think I actually solved some of these in various courses. It's just an absolutely HUGE amount of it. My homework is usually a fifth or sixth of that, which probably makes that course so difficult.
I know harvard professors who say this class is overrated and its better to just take multiple classes that teach the same topics rather than squeeze a bunch of topics in one class.
Those questions actually aren’t that difficult. I’m an engineer and honestly you learn the fundamental rulings it’s all good. Don’t let this intimidate you lol. You can do anything you put your mind too!
I am a graduate mathematics student and I concur. These problems aren’t bad at all. They are problems you’d do in an introductory topology course. Although, if you are not familiar with the material (like most people in this comment section), then it looks like a foreign language.
This is how I felt when my AP Calc teach in highschool handed me Spivak’s first edition of calculus. But it grew me into a wonderful student and love of it. Currently a PhD Candidate!
To everyone here who says that people drop out because of teaching - please don’t judge a book by its cover. Professor Harris has been a great teacher and mentor to me this term in math 55 - his lectures really help you gain a motivation and insight as to why the ideas he introduces are important in the field and we’ve seen some really cool fundamental results from abstract algebra from his lectures. Main reason people drop out after shopping period is simply they don’t want to spend that much time on maths in their freshman year - which of course is not a problem especially if they are doing other hard classes/ comping for clubs etc
In my opinion, no. I personally found 55A to be a bit strange structurally, particularly the multilinear algebra unit where the definitions start to get really dense, but by the unit on Representation Theory you really get a sense of how everything comes together and honestly it’s epic. I must say I am a firm believer that 55B (which I am in rn) is a true crowning jewel of the Harvard maths dept catalogue. We just got out of the first half which is topology and the material is really cool and unexpected: I thought I was going to hate it cus I didn’t like Euclidean geometry all that much but it’s a very different approach to the same field that I found very refreshing. Looking forward to complex analysis though which (according to prof Harris) is amazingly beautiful. As for the ‘50%’ dropout rate that is just pure cap: we actually had a net gain of one student over break, and many of those who joined us this term took physics 16 in the fall which was why they didn’t have the time to juggle that and 55A. (In fact I think the 50% dropout rate more accurately describes math 22)
We had lecture three times a week, an hour and fifteen minutes per lecture. And yes, I am aware of that - 55A covers the equivalent of 3 semester-long courses, whilst 55B covers 2 semester-long courses. As we all know though, 55 is only the beginning - a lot of courses in the catalogue for us to explore that require greater mathematical maturity and go at a faster pace.
That would be true for the entire university of Harvard. There ain’t no way some jock or kid scrolling thro insta or tik tok getting into such a prestigious uni. At best they going community colleges lol.
@@CloneReaperJocks get into prestigious universities bro. This is 2023. One of the smartest kids in my graduating class was on the football team, extremely handsome, was in Honors Society and multiple clubs. Even got a scholarship offer to play university football. There was another guy on the football team who was also in advanced classes. Not all "Jocks" fail at life beyond highschool 😂😂
College: hey look, they gave me an input for omega so I can just solve this one and don't need to write the entire equation. Or ... Hey, question 4 looks like question 10, the equation should be nice and clean to work will in question 11 if they're roughly the same.
I believe this is an Introduction to Mathematical Analysis for most undergrad courses. It's not only in Harvard that's difficult. The overall subject is very challenging because it teaches you a complete new perspective of thinking in math and logic with strict set of rules crammed into 1 semester. It's insufferable class if you're a math or physics major but the rewards are worth it.
As a maths student, not disputing that the course is indeed difficult, but the homework you showed is not the hardest shit ever, the terminology might seem odd for most people but once you know what a metric space is, it becomes quite easy. Not trivial but only not the most I would have suffered, my guess is that this homework is at the very beginning of the course since a lot of problems are solved by directly linking the definitions of isometries and metric spaces
My program has an 80% attrition rate, and it's never about the difficulty of the work, just the amount, it's basically a 9-5 except you're using your brain, not your body and homework extends into after class, for sometimes hours. I've made it this far thankfully, and I only have a year left
@@itsgonnabeanaurfromme the program can only allow at most 15 kids in at a time due to how many lab benches we have per semester, even the later semesters don't have 15 available due to the cost and amount of lab equipment each bench gets, but since COVID the recruiting for the program has gone down significantly and the amount of new students aren't filling up the program anymore.
With all due respect why join, I understand the importance of math and theories but it seems unnecessary. 9-5 no pay on problems that you may rarely find in the real world (you are probably in an occupation that requires this field so this might not apply) And I’m general it’s just stress inducing is there an award? Or something that is beneficial? Again this is just from my outside perspective I congratulate you for taking a class that I know for sure I will not pass
@@hi5yearsago392well someone has to contribute to society academically by writing research papers. Not all can slave away at their jobs and only work for materialistic desires. That responsibility goes to the smart people while the lesser ones do the monotonous 9-5.
@@CloneReaperyou had me until smart and lesser. Lotta stuff wrong with that. Firstly, academic ability is just as much a matter of interest and practice as it is your IQ, if not vastly more so. Secondly, IQ and academic excellence are not the sole metrics by which one measures success or intelligence, and believing it is shows a distinctly myopic view on your part. Thirdly, people with less natural intelligence or academic inclination are not lesser. Aside from the ethical problems with that statement, just because they lack in one area does not mean they cannot excel at many others, or that people who excel academically can achieve the same standards for fields that somebody you would consider “lesser” would. Lastly, strong empathy is positively correlated with high IQ and regular book reading, which suggests you don’t practice what you preach. Perhaps some personal study and rigour would serve you better than UA-cam Shorts?
FYI: this class is just an “accelerated honours” for freshman students. It covers Junior and Senior courses that most Math and Physics students would take.
The problem sheet at the end is actually pretty basic. Everything on the page is standard start of semester curriculum if you go to study math at any university.
@@Tururu134 The class is just requires a huge time commitment. Usually intro classes are easy but moving to systems and then data structures can get complex and difficult. My professor for both those suggested we would spent 30-40 hours a week on work from that class. That's just the homework/lab/assignments. But people do struggle and need a lot more time for certain assignments. If you're coming into the class with some experience already, or knowledge of the language (usually Java) then it's maybe better. My Intro class was in python and the software systems class switched to java, so I had to learn and adapt. As for other stem classes it's the same thing, as you go to higher level classes the difficulty level goes up exponentially and lot of the students can't handle it and decide to switch majors or worse drop out
As an engineering school graduate, I feel sorry for anyone having to take that level of calculus and linear. Looked at the problems and noped out. Spending an evening trying to solve one problem of 10 is not very fun, never again.
You’re fine. I only ever had one professor in my engineering courses (aside from math) teach using all of the formal stuff you learn in discrete mathematics. Most of the class noped the fuck out because the professor was first time teaching us like if we were being presented his thesis. Not to mention you also aren’t expected to study topology or do real/complex analysis. It just takes time to study it and learn, and if you don’t have passion for it or a need for it, don’t bother with it. There’s so many topics to cover in math that as long as you know what you need for whatever you plan to do, you’re fine (exception being a professor, since usually you need to cover a wide variety of topics for that one.)
@@Lamb666 It took me a long time to understand the meaning of "Mathematics is just a tool". You just can't learn to use a tool from scratch, it takes a long time. You see the tool being used and blindly use it initially knowing that "A + tool = B". My professor once told, "Mathematics is just a decoration, 99% is useless junk."
@@rockingttalent3666 Yeah, I can see it being useless junk. Never have I used calculus in my engineering career, yet I took 3 years of the stuff and had to solve some pretty boring problems. Best I've done so far is figure out the elongation of a steel rod to verify my sensor operations.
“Here is this homework assignment it should take you about 60 hours. I need it by tomorrow”
LOL
It’s 4 years in 2 semesters
It’s called
Honors Abstract Algebra and Honors Real and Complex Analysis
Not easy not compulsory
Bad enough 18 yr olds already need 30 year's of work experience for a job and now we need to do 60 hrs of homework in 1 day jeez...
The children: time to invent time travel
They need some Harry Potter witch craft machine
“There’s a class so difficult that 50% of students drop out”
Sounds like most college courses
Bro yes. I’m in college doing a business course, which in my opinion is pretty easy. It’s common sense and taking notes.
Three people dropped out of my class last year alone. We had a class of twelve.
@@whyamigae9666 In almost every stem course 50% drop out is pretty common
So a waste of your money and time?
@@pablofarias8674 yes.
@@briangv
Mines a Btec course I’m not smart enough for Stem courses
Something I’ve notice is that classes considered “hard” usually don’t tell you the background knowledge you should have before taking the class. I find this really prevalent within chemistry since nobody tells you how much a good grasp of physics concepts underly chemistry
Yeah because they want cash so there not going to say fck all man. Thats what happens when you let universities have total control with no locks on them
50% that, and the other 50% they just make it needlessly difficult.
Honestly you don't really need to be fully knowledgeable in physics to understand chemistry. HOWEVER, knowing and truly understanding basic physics concepts and being able to apply them in various means will undoubtedly make chemistry easier to learn, understand, and retain.
We had a class called QCP in high school - quantatative chemistry physics
"we wont teach you these info" --> Hard
My Asian parents would deadass throw me into this class expecting high grades
Bro even the calculator couldn’t help you with that 😂
@@A-Stambro, he IS the calculator 💀
okay then I’ll buy that calculator to do this 25hr homework due In one day 😎
A means average
B means beating
C means committing a crime
D means your dead
F means FAILURE
Edit: more accurate
Son , are you ready to go? Say no and you'll get double belt punishment .
This is a whole new level of "Thanks for Learning with Us"
Oh hello there
idk what that even is . where is it from
www.youtubeuis.com/watch?v=RCMXeapwDseY
Let’s go Brandon
No what do u mean ? It’s not learning ITS MIND HURTING
Good thing I will never have to worry about this class.
Amen to that
Forget class, I don't even have to worry about getting into Harvard
@@skullrazor
Forget harvard, I won't even have to worry about college
@Lucifer wtf
@Banana Chan forget school I’m a crack head now 👸
Me who failed pre algebra: yeah i could take that class ez 😎
Bros gonna fail before he even take the test paper 💀
bros gonna trip on the doorframe and fucking die before he starts the class 😭
@@total_blasphemy probably yeah
I wanna sit in this class for a day just to see the faces the students make
I've looked at the homework and the curriculum for the class and honestly it's not an incredibly difficult class. It's difficult mostly because the students in the class have to completely deconstruct everything they know about mathematics which is a lot harder than straight up just learning maths.
If you want to you could actually create a jerry rigged version of this class on your own lol. For the first semester of the class go through the entirety of a linear algebra and abstract algebra textbook of your choosing (for abstract algebra I've heard great things about Judson's textbook and I really like it) then for the second semester go through all of Michael Spivak's Calculus textbook and congratulations, you've completed math 55.
If you want to make up a lecture for the class then look up "real analysis", there's a playlist by "brightside mathematics" who does a great job at explaining analysis. Watch videos 1-14 of that series and you'll get what's a single lecture in that class for the analysis portion of it I think
*half the students drop out*
So... like litterally every stem course lol?
Yeah exactly. 45 out of the 60 in my microbiology class dropped out less than halfway through. Having a lot of biomedicine under my belt now I can safely say that any relatively hard class will have a dropout rate of atleast 50%
Finnish school system >>>, teach the students instead of dumping them with hw
Except, these are Harvard students 😀
Doesnt seem like a very inclusive atmosphere where young adults are nurtured and taught how to think instead of memorization skills.
@@dilkush_21 I read it
wonder how the teachers feel when grading them
@@dilkush_21 No one care
@@dilkush_21 ok lol
They probably won’t even have to grade any because nobody will finish 💀
@@dilkush_21 i didn't read your profile but that didn't change the fact that you rickrolled me with your name bAcKw4rD$
@@dilkush_21 ok
“im sorry..”
“dont topologize”
- Martin Gardner
I remember my ap math professor walking me through a third of a problem from this in high-school and saying this is what college was all about.
Imagine the final question is 1 + 2, It’d probably be something fun to lift off the stress of doing one of the hardest quizzes you’ve ever done.
fr
I would just overthink it ngl
@zzTIME TRAVEL ..bruh.
Nah the hw is 20 to 60 hrs with hard problems. I ain't getting to the final question.
That's the greatest troll ever...
This makes Calculus look like 1st grade math 💀💀💀💀
This is technically calculus. But yes much harder.
Fr
@@JO-tw9bf ok? what's with the facepalm, that's what they said lmfao
What's calculus
@@Real_DripBacon Math that’s hard af
No wonder Bill Gates droped out💀
😂😂
at this point, they're probably asking you questions that they can't solve LOL
sometimes class is hard because of awful teachers
I highly agree with you😌
trueee
Yup 👍
tru
Nah man you couldn't get Albert Einstein to teach me this shit
A 50% dropout rate just goes to show how poorly the class is taught.
Yes ty beat me to it having a high failure rate isn't something to brag about
There are classes at every university made intentionally to weed people out. You want the best of the best in certain programs so having those courses are essential. 50% drop out rate is pretty normal and good often times
If most people could do any class they enter, then degrees would be meaningless lmao
@@MasterMenahem you don't understand for classes like those you require a test before being allowed in to make sure u have the skillet necessary to learn the info a high failure rate means the teacher is shit and the ones who passed didn't need the teacher to learn this info they could have done so with similar resources and no teacher
Or maybe the course is just hard af and too much of a responsibility🤷♂️
I was about to say then what’s even the point of having a class like that
1. To prove that the discrete metric is indeed a metric, we need to show that it satisfies the three properties: non-negativity, identity of indiscernibles, and triangle inequality.
a) Non-negativity: For any points p and g in X, d(p, g) is either 0 or 1. Since distances are non-negative, this property is satisfied.
b) Identity of indiscernibles: d(p, g) = 0 if and only if p = g. This property holds since the discrete metric assigns a distance of 0 only when two points are equal.
c) Triangle inequality: Let p, g, and h be points in X. Since d(p, g) and d(g, h) are either 0 or 1, the triangle inequality holds.
With the discrete metric, subsets of X are open if and only if they contain each of their points as isolated points. Every subset of X is open since each point in a subset is isolated.
A subset of X is closed if and only if its complement is open. Since every subset is open in X, the only closed subsets of X are the empty set and X itself.
Every subset of X is dense since every point in X is isolated, meaning there are no points in X that are not in any subset.
2. To prove that do is also a metric on X, we need to show that it satisfies the three properties: non-negativity, identity of indiscernibles, and triangle inequality.
a) Non-negativity: For any points x and y in X, d₀(x, y) is defined as d(x, y) / (1 + d(x, y)). Since d(x, y) is non-negative, and the denominator is positive, d₀(x, y) is also non-negative.
b) Identity of indiscernibles: d₀(x, y) = 0 if and only if d(x, y) = 0, which happens if and only if x = y.
c) Triangle inequality: Let x, y, and z be points in X. We have:
d₀(x, y) = d(x, y) / (1 + d(x, y))
d₀(x, z) = d(x, z) / (1 + d(x, z))
d₀(y, z) = d(y, z) / (1 + d(y, z))
Using the triangle inequality for d, we can show that:
d(x, y) ≤ d(x, z) + d(z, y)
Dividing both sides by (1 + d(x, y))(1 + d(x, z))(1 + d(z, y)), we get:
d₀(x, y) ≤ d₀(x, z) + d₀(z, y)
Therefore, the triangle inequality holds for d₀.
A subset of X is open under the metric d if and only if it contains an open ball around each of its points. Since the metric d₀ is defined in terms of d, the same subsets that are open under d are also open under d₀.
The metric spaces (X, d) and (X, d₀) may not be the same as metric spaces, but they have the same open sets.
The metric space (X, d₀) is always bounded since d₀(x, y) is bounded by 1 for any x, y ∈ X, whereas (X, d) may not be bounded.
3. Let's examine the given options for defining a metric on R:
i) d₂(x, y) := √(x - y): This defines the Euclidean metric on R, which satisfies all the properties of a metric.
ii
Just call me the thinker 😗
The problems are pretty much first day measure theory hw😭
🤓
Alright! I understand even less now!
give this man noble prize at the very least. i would believe if he said he was the winner of IMO
Not me being tempted to try and solve the problems knowing damn well I still count using my fingers
My freshman roommate took this and was one of the smartest kids in the class as a freshman. Crazy.
All the math homework on his desk didn’t even have any numbers - it just looked like a different language.
Nickkk
you know your math class is hard when you need to learn a new language
My man. I keep forgetting how smart you are.
Welcome Nick 😩
My toxic trait is thinking I could pass this class with an A
So, you were one of the gifted kids in highschool who didn’t study and got A’s in class?
@@jc8153 even that is a stretch. More like you did that at a school with AP classes only, and you’ll probably make a B in this class at best
As an Asian, this shit’s too easy 🥱
Delusions of grandeur
Dunning Krueger
"Im sorry"
"Dont apologize..."
Bro even the guy making the paper is a statement 😭
"Im sorry, dont topologise" bro decided to put a pun in an excruciating task
"I'm sorry"
"Don't topologize"
mathematicians need some humor in their work or else they're gonna go crazy
what ref is this
@@corncobbob2326lmfao 😂
if I were a professor, I would make up absurd scenarios for math problems. Similar to stuff in "What If" by Randall Munroe.
LMAOO
@@corncobbob2326Topology is one of the classes in math departments.
This is like calculus and algebra combined with honor. I'm good at math when I use calculator if not, I'm screwed.
No. It has undergrad mathematics passed Calculus
So stats squared.
It’s the problem solving and thinking routes that’s so difficult, not necessarily the calculations and arithmetic, so a calculator won’t help you.
It’s like saying am good at making food by ordering food from a restaurant
So your basically your saying you suck at math and need a device for literally everything
And this is where Photomath comes in😂
The fact that there’s an apology at the start of the assignment
Harvard: So what's 2/2
Student (Sweating): Okay so let's start by integrating.....
Possible answer:
a): X AE-Xii
b): X AE-Xii
c): X AE-Xii
d): X AE-Xii
e): X AE-Xii
@@NotSynu bro I think it’s just 1? You did it have to complicate it?
@@Etihad_is_king it just a joke sheeesh
@@NotSynu ok, I didn’t know sorry.
@@NotSynu ou mean elon musks son?
This is what’s wrong with professors these days. They try to make it hard and never really focus on actually teaching you anything.
it’s literally harvard the fuck did you expect? a = 1 now find a?
@@bigchocolate1118 no shit but when you make it to the point where you don’t teach how you got there then you ain’t teaching you just putting out the assignment. You don’t learn that way. I had professors who would just put out the work and the assignments but never really taught. Like how am I suppose to learn that and do the assignment when you didn’t teach us shit smh Had computer science classes that were at the intro level c# class and the professor would never show how he got to the end point. Like mf how did you get there? Like this is an into class and I really need to know that! Please show me got damnnn it 😂
What makes you think the Math55 professor isn't teaching?
@@jmemixx exactly. At that point I'll just drop out and teach myself since the people who are paid ass loads of money to sit and do jack shit and expect us to use sources to learn won't do it. Literally had professor's like that and it pissed me off to no end especially since they're stem courses.
@@bigchocolate1118 That's not the point if they were actually taught to the point of understanding then it wouldn't be half dropping out. they're Harvard students, they are smart- unless half of the class didn't study at all or the least smart in Harvard then the number of dropouts shouldn't be that much unless they're not taught well. Yes, like you said they're Harvard students; do you expect them to be dumb and not study at all? Lol, you think half of the class are 'shit students'? Well, I don't think Harvard would accept that many 'shit students'?
In my opinion, having a class like this is ridiculous and not a flex. I’m not sure how it is in America, but I’m Australia each unit should expect about 10 hours (including contact hours) of work weekly because 4 units makes a full time study load. That unit better be worth 4 times the credit for it to be worth taking.
And I was literally out here struggling with grade 8 math💀
me too 🥲
yea like broo literally...
@@linalovesfolklore i love ur profile smm
Yeh fr💀
PLEASE LMAO I HAVE MY MATH FINALS TOMMOROW LIKE IDK SHIT😭😭
Saw the word “math” and I’m already dropping out 😂
Math is atleast more bearable than fine arts, also math majors like engineering make more money.
@@clashoclan3371 because maths is more hard of course u get more money
@@Starlight_is_cool math isnt more hard tho. Fine arts are. The only reason they are easy to pass is because of what also makes them hard.
Like languages, they arent logical or structured at all. But that also allows you to pass things like art.
Math is super easy because its 100% logical so it makes total sense. Nothing just “is” without a reason(or at least one that is a bit more advanced than your level)
@@o_sch Okay but consider that it maybe easy for someone like you because you need logic and structure, whereas for others, fine arts is easier because it doesn't have a definite answers for the most part.
@@Thewritingelf thats probably true but objectively definite answers are the best for development and advancement of society/technology.
Ooh metric topology was one of my favourites in undergrad! Those first two questions are pretty solid ones too, not too hard but definitely need a good understanding.
Ngl that looks pretty easy (the only points I’ll get it for my name)
Well, this just gives me flashbacks to today and everyday for the next 2 years... cause a lil brat thought "oh, I like maths, so why don't I pick my nations hardest most intense maths program even though I hate homework"🥰
Oof
You got targeted dude
Daym so many sus pages
Damn all the bots just decided to jump on this one comment, I'm sorry for you random math enthusiast
@@beleques 🥲
Now this really makes me wonder. When would they use that knowledge in the real world
It’s more to flex how smart you are than to actually use it in real life
Right? Even im confused
Ikr
Jobs?
Math gets really fucking hard. If you want something relevant to you, please look at how rendering engines calculate light paths, and enjoy the math rabbit hole.
“I sentence you to a full semester in math 55”
Tbh Ive done those concepts and I passed each course that taught each topic with a really good grade but I can see how ppl might find it hard, it’s a lot, and also a lot of problems. I’ve had a similar class that the professor gave 1 assignment each week that was 15 double sided pages it was for a discrete math course it was so much work that I ended up dropping another class so I can put more time into that class
honestly, most math grad can do those questions. The course is challenging because it’s an intro course that compacts ~5 course topics. However, it’s not difficult overall
As an engineer (not even as a math grad), I agree 😊
And this is why I am not and never will be a math grad. 👀
Just looking at this assignment gives me PTSD...
It may not be particularly difficult, but going into a maths course expecting highschool 'algebra', you will be sorely mistaken.
Nerd
@@keylan9191 mad?
"Im sorry"
"Don't topologize"
- Martin Gardner (adapted)
Lies again? Shin Tokyo
@@ayoromanchik (adopted)
“Be better” -Kratos
lol
My brain ain't braining anymore after seeing the questions
"take a look at these questions"
_laughs in potato quality_
i answered math olympiad questions from 4-5th grade.
@@blzbbqheelx1163 cool story bro
@@blzbbqheelx1163 bro I am literally struggling with grade 10 mafs😭 any tips on how to improve?
@@emiyakiritsugu9408 he just capping. only improvement technique u need to know is practice. that‘s it
I am the type of person who will take course just to see how hard it is and since I hate dropping courses, will stick with it. And well probably fail it since I am an idiot who took it even when others warned me.
Thats what id do actually, and get flunked at the first week or month and still go on
😂
Lol that's what I did! I even dragged my friend in to it. She quit half way, but my stubborn ass wasn't gonna give up and I luckily passed.
Don't take school to heart take take those grades to heart this school system wants you to hate yourself the whole reason school was made is to train you to work like a slave they want you to work for other people instead of working for yourself the guy who created the school system said he wants a world of workers not thinkers they want us to physically do hard labor work for money even tho we get paid nothing school is useless unless your working towards a job that requires school then do that but if your job requires no school then you don't need school most of the things school teaches you is random bs that we will never use in the future 90% of the time like history they teach us meaning less things to just distract us the teachers even know it they just do it for money and if you wanna leave school or try to drop out you will get involved with the court and you or your parents could get jail time even tho you don't need a diploma if you know you got the mentality to do your dreams do it just don't doubt yourself
@@yunggrouch.9659 i agree with u but ur grammar and wording seems like u need to go back to school again
1 = The discrete metric is a metric. All subsets of X are open, all subsets of X are closed, and all subsets of X are dense
2 = The metric space (X,d0) is always bounded
3 = The metrics that define a metric on R are ii) d2 (x,y, and v) d5 (x,y)
4 = The given statements i), ii), and iii) are proven. Two metric spaces (X,d) and (X, d0) are isometric when there exists a bijective isometry between them
When I was a kid I decided to major in math because geometry and computation were fun. Only thing I was good at. By the time I hit Mathematical Analysis in college (a math class where your TYPED homework answers with symbols looks like essays 🙂) it was too late to switch majors. That’s where villains are made, but I still earned my degree all the same~* if you’re a student mathematician at ANY level, keep pushing!!!!! Much like anywhere else in life, only the persistent last.
they call it math 55 because that’s the average grade for people who take the class
thats still quite high tbh, most biochem courses are like 40
that would make sense tbh
@@javier.a.vargas
Well, if half the class drops it, it’s safe to assume only the higher performing students remain
@@guccifer764 indeed in my case we were 50 students and only 17 finished the course, the class average was 57 , biochemistry 2 analytical spectroscopy and gaz based ccm
And lemme guess, the final is a free for all showdown and whoever wins is declared king of math
"All hail the new Math King! 👑"
Wth with the bots bruh
@@mmmeli6296 they vibin
bro… whos after you?
Bots just love me don't they
I put some of the equations into a math solving ai tool and even the ai couldnt solve it 💀
Because it can only really spit back information, not actually process it in a conceptualized way, yet. That would require an Ai with self learning and recognition. That is a few levels above chatGPT or whatever u are using. 😅
@@cielonehellofaservicedog4648still though, the fact a Ai that has access to a lot of human knowledge, yet still can’t figure this out is funny to me at least.
These problems really aren't that difficult. They are just university-level math. Not easy, but not impossible either. I'm not good at english math, but I think I actually solved some of these in various courses. It's just an absolutely HUGE amount of it. My homework is usually a fifth or sixth of that, which probably makes that course so difficult.
The amount of people not realizing this is a joke is insane
@@evanrutledge-sz4yoai doesn't really have infinite knowledge lol
At the moment ai is still very bad at math
I know harvard professors who say this class is overrated and its better to just take multiple classes that teach the same topics rather than squeeze a bunch of topics in one class.
Those questions actually aren’t that difficult. I’m an engineer and honestly you learn the fundamental rulings it’s all good.
Don’t let this intimidate you lol. You can do anything you put your mind too!
Go on, solve em
I am a graduate mathematics student and I concur. These problems aren’t bad at all. They are problems you’d do in an introductory topology course. Although, if you are not familiar with the material (like most people in this comment section), then it looks like a foreign language.
give us the answers
@@LeonhardEulerShades then solve it
Solve it
" the class is called MATH 55"
*Scrolls*
then how'd you comment?
@@snail123O *_Scrolls to the comment section..._*
Like a failure rate is something to brag about. It just shows that the teachers suck at their jobs, or that the instruction material sucks ass.
took one glance and said “hell naw”
This is how I felt when my AP Calc teach in highschool handed me Spivak’s first edition of calculus. But it grew me into a wonderful student and love of it. Currently a PhD Candidate!
It reminds me of the *_mold in this video_* ua-cam.com/video/cJpn0wkihWk/v-deo.html&.njira
@Atman Gotango ayo!? What u talking about?
Phd candidate for what?
"I am sorry"
.."don't topolgize'
I am dying😭😭
Heehee
This kind of looks like my upper level abstract algebra class with some real analysis thrown in there
Luckily for us we only need 1 person that passed to put the answers on the internet
To everyone here who says that people drop out because of teaching - please don’t judge a book by its cover. Professor Harris has been a great teacher and mentor to me this term in math 55 - his lectures really help you gain a motivation and insight as to why the ideas he introduces are important in the field and we’ve seen some really cool fundamental results from abstract algebra from his lectures. Main reason people drop out after shopping period is simply they don’t want to spend that much time on maths in their freshman year - which of course is not a problem especially if they are doing other hard classes/ comping for clubs etc
Is this class really as hard as gohar is making it out to be
In my opinion, no. I personally found 55A to be a bit strange structurally, particularly the multilinear algebra unit where the definitions start to get really dense, but by the unit on Representation Theory you really get a sense of how everything comes together and honestly it’s epic. I must say I am a firm believer that 55B (which I am in rn) is a true crowning jewel of the Harvard maths dept catalogue. We just got out of the first half which is topology and the material is really cool and unexpected: I thought I was going to hate it cus I didn’t like Euclidean geometry all that much but it’s a very different approach to the same field that I found very refreshing. Looking forward to complex analysis though which (according to prof Harris) is amazingly beautiful. As for the ‘50%’ dropout rate that is just pure cap: we actually had a net gain of one student over break, and many of those who joined us this term took physics 16 in the fall which was why they didn’t have the time to juggle that and 55A. (In fact I think the 50% dropout rate more accurately describes math 22)
@@gregwong-dg5jq how many hours a week do you have lecture? Those topics you are naming, typically take a whole semester
Hows class going?
We had lecture three times a week, an hour and fifteen minutes per lecture. And yes, I am aware of that - 55A covers the equivalent of 3 semester-long courses, whilst 55B covers 2 semester-long courses. As we all know though, 55 is only the beginning - a lot of courses in the catalogue for us to explore that require greater mathematical maturity and go at a faster pace.
And here I am suffering in algebra 1
Bruh me too I could never
Exactly. I barely understand it
It’s pretty easy when you get the hang of it. What are you doingv
Yeah I finished two years early last year, seventh grade… it was sooo hard ;-;
@@mihirnike001 The math, poorly
Having half your students drop sounds like something you shouldn’t be proud of as a teacher 🥶
I’m having a panic attack just looking at the questions
You can walk in that class, scream "Yo nerd" and everyone would turn their heads
That would be true for the entire university of Harvard. There ain’t no way some jock or kid scrolling thro insta or tik tok getting into such a prestigious uni. At best they going community colleges lol.
@@CloneReaper we watching a yt short rn bro its worse
@@CloneReaperJocks get into prestigious universities bro. This is 2023. One of the smartest kids in my graduating class was on the football team, extremely handsome, was in Honors Society and multiple clubs. Even got a scholarship offer to play university football. There was another guy on the football team who was also in advanced classes. Not all "Jocks" fail at life beyond highschool 😂😂
@@CloneReaper kid who can throw a ball very far:
@@CloneReaperyou are disconnected from reality if you think kids at prestigious universities don’t use tiktok or insta
Elementary me: "I FOUND A LETTER IN A MATH QUESTION!"
High school me: "I FOUND A NUMBER IN A MATH QUESTION!"
College: hey look, they gave me an input for omega so I can just solve this one and don't need to write the entire equation.
Or ...
Hey, question 4 looks like question 10, the equation should be nice and clean to work will in question 11 if they're roughly the same.
Lol our math class in our first semester had a failure rate of 75%
Classes are so easy just pay attention
I believe this is an Introduction to Mathematical Analysis for most undergrad courses. It's not only in Harvard that's difficult. The overall subject is very challenging because it teaches you a complete new perspective of thinking in math and logic with strict set of rules crammed into 1 semester. It's insufferable class if you're a math or physics major but the rewards are worth it.
Harvard has a freshman analysis class which is easier than this.
Agreed. I took complex analysis and it was very rigorous. I'm an applied maths major btw. The difference is, I'm on quarter system 😭
@@tacosamuel math55 is:
Honors Abstract Algebra and Honors Real and Complex Analysis
So this is 4 years math in 2 semesters
2 semesters are here
Im sorry only an introduction?
As a maths student, not disputing that the course is indeed difficult, but the homework you showed is not the hardest shit ever, the terminology might seem odd for most people but once you know what a metric space is, it becomes quite easy. Not trivial but only not the most I would have suffered, my guess is that this homework is at the very beginning of the course since a lot of problems are solved by directly linking the definitions of isometries and metric spaces
It does say homework assignment 1 but i agree that this isnt particularly hard
This
Exactly, is not that I am the most talented math student, I think this is an average introduction homework. Great point of view 😃
I was thinking the exact same
Its a first year maths course tho, If someone gave me that assignment on the first week I would have dropped out 🤣🤣🤣
“yeah im sticking w/ chemistry”
Now chat gpt gonna make that class a cakewalk
My program has an 80% attrition rate, and it's never about the difficulty of the work, just the amount, it's basically a 9-5 except you're using your brain, not your body and homework extends into after class, for sometimes hours. I've made it this far thankfully, and I only have a year left
Easy to have a high attrition rate when you have low standards for accepting students in the first place. Just saying.
@@itsgonnabeanaurfromme the program can only allow at most 15 kids in at a time due to how many lab benches we have per semester, even the later semesters don't have 15 available due to the cost and amount of lab equipment each bench gets, but since COVID the recruiting for the program has gone down significantly and the amount of new students aren't filling up the program anymore.
With all due respect why join, I understand the importance of math and theories but it seems unnecessary. 9-5 no pay on problems that you may rarely find in the real world (you are probably in an occupation that requires this field so this might not apply)
And I’m general it’s just stress inducing is there an award? Or something that is beneficial?
Again this is just from my outside perspective I congratulate you for taking a class that I know for sure I will not pass
@@hi5yearsago392well someone has to contribute to society academically by writing research papers. Not all can slave away at their jobs and only work for materialistic desires. That responsibility goes to the smart people while the lesser ones do the monotonous 9-5.
@@CloneReaperyou had me until smart and lesser. Lotta stuff wrong with that. Firstly, academic ability is just as much a matter of interest and practice as it is your IQ, if not vastly more so. Secondly, IQ and academic excellence are not the sole metrics by which one measures success or intelligence, and believing it is shows a distinctly myopic view on your part. Thirdly, people with less natural intelligence or academic inclination are not lesser. Aside from the ethical problems with that statement, just because they lack in one area does not mean they cannot excel at many others, or that people who excel academically can achieve the same standards for fields that somebody you would consider “lesser” would. Lastly, strong empathy is positively correlated with high IQ and regular book reading, which suggests you don’t practice what you preach. Perhaps some personal study and rigour would serve you better than UA-cam Shorts?
okay let’s take a moment to appreciate the fact that atleast half of the students suffer the full semester 😃
Yay i’m the only non bot here
@@magmafeather so am i
@@magmafeather i am here too lol
💯👹ua-cam.com/video/LGzz9ql-JK8/v-deo.html
I'm literally drowning in these links
You know it’s bad when it has more letters than numbers
Me when I finally solved 1+1: 🎉😊🎉🎉😊😊🎉😊🎉😊
Me when I see this question:
Bruh, forget the class, I felt like dropping out of consciousness after I saw that sheet.
Better to just be homeless
👹💯ua-cam.com/video/LGzz9ql-JK8/v-deo.html
Teacher: The test isnt that hard.
The test: Jimmy has three apples, how many atoms are there in Saturday on Wednesday weekend.
💯👹ua-cam.com/video/LGzz9ql-JK8/v-deo.html
There are 4 apple atoms on Saturday Wednesday weekend,
@@Random-mm2ly Did yo find the standard deviation of F(x) = mx+b < 7!^3? Cause if not, it might be 5.
@@money0925 uh wtf I'm still im middle school pls teach me
@@money0925 uh wtf I'm still im middle school pls teach me
My ego would force me to stay in the class, whether my downfall occurs or not.
The homework presented is actually pretty easy nomenclature and initiation requisites aside.
" I'm sorry.."
"Don't topologize"
- Martin Gardner
"And I took that personally"
- Math
I didn‘t get it till i saw it on the math examples lol
Topography humor I rekon
It reminds me of the *_mold in this video_* ua-cam.com/video/cJpn0wkihWk/v-deo.html&.rjaba
Looks like we'll never figure out how many apples Johnny have in his pocket.
He has infinite amounts of apples
Made it 69 likes
*_NOICEE_*
I swear, these bots are getting on my nerves.
Bot paradise
👹💯ua-cam.com/video/LGzz9ql-JK8/v-deo.html
At my university the first math module has also like a 50% failing quote
Honestly these homeworks look just like every university math homework...
Fun fact: Simpsons executive producer Al Jean also took this class.
Scary fact not fun
ic...
thats why simpsons had so many 'coincidental' future predictions...
sus👀
@@creativeastronauts6894 the parents watch the news and the children watch the simp sons but its all the same.
👹💯ua-cam.com/video/LGzz9ql-JK8/v-deo.html
Conan was also a writer for the simpsons and went to Harvard
Me: can't be that hard
Guy: the class is called math-
Me: damn ok he was serious 👀
pafp
hahaha righttt
lmao
Math ez
@@foden4420 -4th grade kid who's learning fractions
FYI: this class is just an “accelerated honours” for freshman students. It covers Junior and Senior courses that most Math and Physics students would take.
Dear brocolli, that is no longer math, that is wizardry
When you finish an assignment in this class but then you realize you used a different formula than the teachers.
Your profile picture:
@@-aftermath-8508 that’s what I was gonna say 😆
@@crispycoochie7760 SAME
Omfg i just did this last night it was an summative test and holy crap i used the wrong formula and half of my answer sheets are wrong
👹💯ua-cam.com/video/LGzz9ql-JK8/v-deo.html
It’s doable, some people still get A’s in this course, it’s just that there will be sacrifices
Human sacrifices
It reminds me of the *_mold in this video_* ua-cam.com/video/cJpn0wkihWk/v-deo.html&.bvxra
@@bimbam0000 Satan controls Harvard
Sacrifice woohoo
This assignment to take 60 hours listen, Google exists And Google takes five minutes
Take a look at the question you won’t use for life
When you pause the video and finish the problems at the end
Just to realise it’s been 7 years
Lol
what year is it ?
it’s 2028 boy, you spent too long on that equation
" U try it."
Oh hell nah
My toxic trait is thinking I can do this class 💀💀
The problem sheet at the end is actually pretty basic.
Everything on the page is standard start of semester curriculum if you go to study math at any university.
Me as an overconfident 20 year old who dropped out of college after one year: "Math? I could get an A in that class easy."
This is true for literally every single stem class.
Computer science especially. First data structures class is down by 50%+ every single semester
why?
@@Tururu134 The class is just requires a huge time commitment. Usually intro classes are easy but moving to systems and then data structures can get complex and difficult. My professor for both those suggested we would spent 30-40 hours a week on work from that class. That's just the homework/lab/assignments. But people do struggle and need a lot more time for certain assignments. If you're coming into the class with some experience already, or knowledge of the language (usually Java) then it's maybe better. My Intro class was in python and the software systems class switched to java, so I had to learn and adapt. As for other stem classes it's the same thing, as you go to higher level classes the difficulty level goes up exponentially and lot of the students can't handle it and decide to switch majors or worse drop out
It reminds me of the *_mold in this video_* ua-cam.com/video/cJpn0wkihWk/v-deo.html&.iuzua
@@geeet_eth i chose CS as my major and i never had CS classes so i really dont have any idea of how its going to be but i do know some simple stuff
This was explained by spencer reid in criminal minds
I'm 101% sure i'm gonna fail that class in the first day if I get into it
As an engineering school graduate, I feel sorry for anyone having to take that level of calculus and linear. Looked at the problems and noped out. Spending an evening trying to solve one problem of 10 is not very fun, never again.
You’re fine. I only ever had one professor in my engineering courses (aside from math) teach using all of the formal stuff you learn in discrete mathematics. Most of the class noped the fuck out because the professor was first time teaching us like if we were being presented his thesis. Not to mention you also aren’t expected to study topology or do real/complex analysis. It just takes time to study it and learn, and if you don’t have passion for it or a need for it, don’t bother with it. There’s so many topics to cover in math that as long as you know what you need for whatever you plan to do, you’re fine (exception being a professor, since usually you need to cover a wide variety of topics for that one.)
@@Lamb666 It took me a long time to understand the meaning of "Mathematics is just a tool". You just can't learn to use a tool from scratch, it takes a long time. You see the tool being used and blindly use it initially knowing that "A + tool = B".
My professor once told, "Mathematics is just a decoration, 99% is useless junk."
@@rockingttalent3666 Yeah, I can see it being useless junk. Never have I used calculus in my engineering career, yet I took 3 years of the stuff and had to solve some pretty boring problems. Best I've done so far is figure out the elongation of a steel rod to verify my sensor operations.
Read through the syllabus and homework, seems like a pretty standard first year abstract math course
It reminds me of the *_mold in this video_* ua-cam.com/video/cJpn0wkihWk/v-deo.html&.vaswa