The (very) basics of sets part one. -- Proof Writing 1

[@abishekpuri5684]

[20:36] The cartesian product of {1,2,3} and {-1,0,1} doesn't contain (0,0), (0,1) or (0,-1). I think you need to shift all the dots one value to the right :)

[@marius3023]

This is the best series you could make. I absolutely love it. ❤️

[@n0mad385]

Frrr I wanted to self study this topic but its so monotonous reading books, even ones (like the one referenced at the beginning of the video) knowing damn well I have to study for my actual field of study.

[@officiallyaninja]

I've been wanting to study proof writing for some time now! I've been trying to learn real analysis and algebra and I feel like my lack of experience writing proofs is really holding me back, and I'm an engineering major learning math in my free time so it's not like I have a math prof to help me.

[@mohamedaminehadji6415]

If you have math courses in your program, they are very probably taught by mathematicians (familiar with proof writing). They can definitely help you with that

[@Debg91]

Back when I was taking calculus in my first year of uni, we had a not so good professor who expected us to write proofs without any previous teaching on the subject. This would have come in very useful, thank you.

[@TooManyKuromosomes]

a venti pfp under a math video :o

[@mitchellyolevsky4376]

Hi Professor Penn, I've recently started studying math privately as an age 40 something adult with the aid of an awesome individual who is a doctorate candidate and T.A. at the University of Toronto Department of Mathematics. We're currently working through a book called A Readable Introduction To Real Mathematics by Daniel, David and Peter Rosenthal. The content you put out is tremendously helpful as a supplement to the book we're working through. I'm not a very skilled proof writer, but I wanted to thank you for sharing helpful material that benefits people like me who aspire to become more skilled at the craft of mathematical proof writing.

[@ikhu6042]

14:30 implying pie is a vegetable. jokes aside this is a really good video

[@ouni2022]

Thanks! Happy new year Mr Michael!😊

[@WillhelmLiebniz]

I just got a paperback copy of this book and found this channel. I'm so excited.

[@OnionBread-41]

I am taking my first proof writings course next semester, so this was excellently timed!

[@sillymel]

(14:53) Pie isn't a vegetable, Michael. (16:50) The x-axis is a *subset* of that set, not a member. Also, you haven't defined subset yet. (20:32) That's {0, 1, 2} × {-1, 0, 1}, not {1, 2, 3} × {-1, 0, 1}.

[@maldi_tof2910]

That really looks more like (pie)/4 than it does (pie) LOL

[@bobh6728]

-3 points, so I’d give him a B+ for a grade!

[@squarerootofpi]

Mum says I should eat more vegetables. I ought to show her this video.

[@ALEXANDERCHEWTIANYOUMoe]

He didn’t say that the second object in the tuple is a vegetable

[@maldi_tof2910]

I don't know why I am so excited to see this. 🥰

[@ranakmadan3010]

omg im so glad i subscribed i am literally learning how to structure number theoretical proofs

[@dylonlarue8350]

Outstanding book, I’m glad you have a series for it!

[@Ny0s]

The book of Proof is a really nice book, I'm enjoying the idea of this series. Thank you very much for the work, I'll watch it all!

[@BethKjos]

Minor disputation: There is no computable upper bound on the number of words in English. There are finitely many words attested on paper at any given moment in history, but it is undecidable whether the language overall admits a finite vocabulary of words which, if pressed, could be made to serve the English tongue.

[@Kycilak]

There is surely an upper bound on the number of letters in a single word. We can take the maximum information that the interactable universe translated to a word length as the maximum word length. Then by combinatorics we get the upper bound on the number of possible words hence it is finite as English is constrained by our reality (through its users).

[@BethKjos]

@@Kycilak Ok if you want to get pedantic... Surely English will never have attested a googol of words for practical reasons, but to place a theoretical upper bound requires making a mathematical argument based on the language definition, which is (in practice) constantly evolving.

[@Kycilak]

@@BethKjos Of course English can have arbitrarily large words and arbitrarily many of them if we are not constrained by reality. But whether you want to call the maximum information the universe can hold a practical reason or not changes nothing about the possibility to make a finite set of all words that can fit into available information of which all written form of English are subsets. Even then when English dies there will be no more mutations and a set of all words ever used in English may be composed then. So yes, there is no constraint in English of how large its dictionary can get but its dictionary will ever be finite.

[@klevisimeri607]

I just find motivation every time I watch your videos.

[@robin1826]

Thank you!

[@Nikolas_Davis]

Great, a new series! 😀 btw, the "vegetables in your fridge" set is not well defined if you happen to have tomatoes in your fridge ;-)

[@Galagaler]

answers to the warmup questions????

[@knowyourworld9605]

22:00 english dictionary malfunction. Please hold 😂. Great video btw

[@pfever]

Amazing course! and unique material on youtube! I really appreciate this! I am an engineer working with artificial intelligence and to advance in research I believe I need to understand mathematics at a deeper level and I think this course will allow me to do that, thank you!

[@Minor_meng]

wow thanks you so much, it's definitely a very useful video

[@abdurrahmanlabib916]

Is the differential eq platlist compelete?

[@HarshilHandoo-cm5gk]

This is like my discrete structure course but just with the proofs part

[@kapkarakoyun]

Thanks I try to learn by myself

[@simianomatlaldo9022]

Any idea about what to do with the third example in 1 and 2 at the end?

[@plislegalineu3005]

Average American thinking a pie is a vegetable 14:50

[@CTJ2619]

wow a course on proof writing

[@bibekpaul420]

Phi belongs to every set,so,does every set of n elements have n+1 elements? Time 8.00

[@soloanch]

Phi is a subset of every set is what he meant.

[@TheAlbert18361]

Warm up exercises 1c is 1,2 2,3 1,3? 2a is the set of all y=x+c where c is all integers?

[@TheAlbert18361]

2c surface of cyclinder?

[@ayrthhhn]

wow thanks

[@THEMATT222]

Noice 👍 Thanks 👍

[@moshecallen]

Well done but I would have noted that cardinality generally denotes the number of distinct elements. I can write a set which is a countably infinite set of 5's, but by the definition of cardinality that I know of, that would be a set of cardinality 1 until and unless I introduce some manner of distinguishing the elements such as the set of all algebraically defined ways of representing the number 5.

[@kevinsilberberg9684]

I am confused why in your first example of set builder notation you had the mathematical expression on the left side (2n) and then x in Z on the right side when every other example had the opposite. see {2n | x in Z} vs {x in R | x^2 = 2}

[@Joy-ju8bj]

Any reference book..for proof writing

[@michaellink1218]

What sections of Hammack does this correspond to?

[@Happy_Abe]

That last one is gonna be a cylinder

[@khayalaliyev3519]

5 and 🥦; Professor,You are a different level of mathematician, a math lover from Azerbaijan 🇦🇿;if mathematics were a game,you would be the one of the final bosses

[@benjaminpedersen9548]

What I found lacking in my Mathematics education was a more systematic teaching on how to present mathematical ideas and arguments in a reader friendly way. One or two professors may have shared a thought on the matter, but it was generally expected to be learned by osmosis. For example, there can be a tendency to take the reader by the fastest route, but they can often have a hard time following and seeing the need for a subargument if they haven't been presented with the overall structure of the argument.

[@antonyjames4441]

R x Z is right??

[@CTJ2619]

ready, SET, go

[@khaledchatah3425]

The weird thing is: A set is a collection of elements Then why do we call the empty set a set? Since the empty set is not a collection of elements cuz we have no elements to be collected

[@sillymel]

"A set is a collection of objects known as elements" is a pretty crappy definition, in all honesty. (No offense to Michael.) Granted, it's actually pretty hard to properly define a set without introducing contradictions, so the actual definition of a set is almost certainly way outside the scope of this course. To vastly oversimplify things, though, the definition works kind of like this: 1. The empty set is a set. 2. If you have some things (or just one thing) that you know are sets, you can manipulate them in certain ways to get new sets. (The complicated part of the definition of a set is the part that tells you how you can combine pre-existing sets to get new sets.

[@StaticBlaster]

0 is not in N because in the natural world, there's always at least one instance of anything.

[@tinnguyen2031]

when are the daily videos on the other channel coming back?

[@TommyFink-y6c]

Walker Lisa Miller Timothy Moore Steven

[@khayalaliyev3519]

Maybe, sometimes you forget English 😅

[@edmundgerald5764]

Been watching your videos for sometime now. Please edit out freezes and scratchings - make yourself more professional and credible. Further, if possible, speak a little faster.

[@BboyKeny]

I like the authenticity and rawness of these videos. The credibility for me comes from the fact that the math is correct. Maybe it's a cultural and/or generational difference. I'm from the Netherlands and 29 years old. I watch all video's except for music videos on 2x speed, I so can't comment on the speed. My only unsolicited advice would be that he shouldn't be too hard on himself whenever he messes up his speech, since I can see him visibly upset whenever it happens. But it very human, and beating yourself up over it only increases the freezes and scratching since it increases the stress. But being self conscious about these "ticks" also increases the stress. Conmen are fluent, professional and credible looking people, since that's their whole shtick. I think that's why I prefer people that you can see actually trying to do their best, instead of jump cuts till perfection. I'd rather have more and regular content from a mathematician, than a few video's from someone that's unmotivated because they have to become a full time video editor making all the other parts of the content suffer. But too each their own :D

[@edmundgerald5764]

@@BboyKeny Noted.

[@chrisjuravich3398]

Some of us are just learning this or seeing it for the first time. Slower speech is much appreciated. There are plenty of fast-talking math guys on other channels.

[@edmundgerald5764]

@@chrisjuravich3398 noted