I had you at U of T for MAT137! I had a lot on my plate at the time so I didnt do very well, but Im back at school now and my Major is Mathematics because of your course. You're the most memorable prof I've ever had. Still coming back for your lessons. Thank you very much!
Woah! I have a lot of mixed feelings about that class, if you didn't know the coordinator for 137 that year Alfonso tragically passed away. But regardless nice to see you again and congrats on finding your way back to math!
Scaffolding Thinking! Clarity! A 68 yr old woman, not a mathematics major, minor, medium - 'your name' caught my attention 'once again' after two years of whooped-dog-feeling avoidance (due to so much rewiring occurring) of completing an online mathematical thinking course. Spent the morning rewinding your video between numerous other brief interruptions. Invaluable! Keep 'em coming. Returning to Art, persisting at Spanish, and squeezing forward a row to reclaim that seat in math class, yet... this Rural Elder Thrives! Number Theory seems to entice, too. Thank You. PS I recognize the shifts in temple hair (not beard) color, all too well! Chuckle.
Thank you for this. I think you make the same point that I came up with after failing my 1st Linear Algebra test. My Calculus 3 course is all calculations and execution of Algorithms which I'm doing fine in and didn't realize I was so used to them that I was not as familiar with the theorems or proofs. My linear algebra course blind sided me because the professor went heavy on the test with theorems for proving if something is always, sometimes, or never true when backing up your statement. I will apply your process for every definition I see to make sure I have a firm grasp of the definition and theorems
6:28 The "write down the definitions" method works *really* well IME for set theory and topology theorems! Like: "Prove that if A \int B = A then A \subset B" - writing A is a subset of B as "if a is in A, then a is in B" and write A intersection B as the set if elements x such that x is in A and x is in B, the theorem falls out immediately!
Thanks for this video! It really made a lot of concepts clearer for me. Proofs have this fascinating property that sometimes they seem really arbitrary for me and I have no idea how someone can come up with that, but it’s nice to know that there is some sort of scaffold that you can use to rise to the conclusion!
I have my final exams next week and i am so worried my whole geometry exam is based on proofs like for eg sss sas asa and converse of isos triangle theorem, isos triangle theorem and way moreeee
Long story short: the information in this video is great.^^ When I went through this, I realized that I knew almost everything that Dr. Trefor is talking about. This video puts together all information nicely, that I felt the "dots are connected" :)
I had to take a discrete math class once, and noticed a common issue that trips me up is remembering the definitions of things. For example in the case of "If X is even then X squared is even." I forgot what the actual definition of an even number was lol.
Professor I am a high school student from india preparing for my engineering entrance exam and want to understand maths to it's core and so I often search out mathematical informative videos on the internet apart from studying the regular course and this is the best video I found over the domain I worked upon, it just made me think to depth, it was a really simple video with a lot of knowledge being given Thanks a lot
I'm from India .where we have to cover huge sllybus may be in just 2 months of one semster and my tution teacher also give me practical eg in maths ie you give about wife and dishes . But these eg are really helpful to learn maths
Also I think it’s worth noting to write down the exact definition, e.g. something says 2 curves are tangent, define that as having a line that is the common tangent to each curve, rather than saying the two curves touch one another
When I taught myself Calculus I used to come up with my own derivatives and integrals, meaning I would throw in random function and try to diff/int them. So my question to you is can I apply this in terms of thinking up a random proposition and then try to show if it holds or not? I know I'll have to be precise with definitions and my logical steps but pretending were in fairyland where the rules are loose, is this possible? I'm asking this because if I run out of exercises I want to see if I can state some proposition and see where it takes me
Hello Prof, Thank you for this video. This is more like a light in the tunnel. However, this whole real analysis stuff looks totally strange and kind of challenging to me. Possible textbooks or links to solidified my understanding will be appreciated 👍. Thank you Professor
Thank you for this video. I double majored undergraduate in physics and EE, the physics was focused on particles, and I’ve always felt weak in proof. The idea of playing around with a proof is revelatory, and some areas of proof come easier-perhaps that’s tied to my understanding of certain areas. I’ve had a block with proof in probability theory, but not with topology-perhaps it’s because I can visualize (referencing back to particles and fields) an analogy to “see” the math problem referenced, better. This video has expanded my understanding thank you. Is there a book you’d recommend for reading relative to mathematical proof?
I've found "How to Prove It: A Structured Approach", by Daniel Velleman to be quite good. It goes into detail on quantifiers, set theory, and proof techniques (induction, contrapositive, etc.)
so what I am understanding is that which proof method is best in which situation is just trial and error of trying each one see if it leads anywere if not try using the next method to proov is that correct?
I love this video and it really helped in my proofs, but ive found this question stating prove that x^3 -x is a multiple of 6 I know the answer now but how should I've tackled this question
How do we use different proofs to prove the negation of cubic equation and its given negations of its x values? All the examples shows for odd and even intergers only.
Thanks for doing this video. I am confused as to how you defined even numbers in your assumption as x=2p, where p is an element of Integers. Aren't even numbers supposed to be numbers that are devisable by 2 with zero remainder? And also, it is again weird for me that the conclusion is X squared = 2q, where q is an element of integers. Theorems are such a weird thing that make no sense to me regardless of how long I stare at them and try to figure them out. Extremely frustrating to be honest. Edit: Actually it's making sense to me now haha :D , the integer definition part I mean.
@@DrTrefor Jokes aside, your channel is great and always helps me to clarify any doubts that I may have. Greetings from Argentina and keep up the great work!
Professor, thank you so much for posting this video. I am trying to learn Proofs on my own, and the breakdown of how to study and do proofs is very definitely interesting and has practical value as well, esp the tip suggesting to come up with Geometrical pictures and the one where you ask to come up with concrete example that show that the proof holds. So there is both theoretical and practical value in what you are asking us to do. Will definitely take a few proofs and apply your steps to it. Thank you very much once again for a quality video.
I'm not sure if this has been stated clearly in the vid, but the "do your manipulations" phrase means: proceed according to the laws of logic. The laws of logic have this wonderful property that they always lead from a true assumption to a true conclusion. ALWAYS. The laws of logic are called tautologies. If you use these to draw conclusions repeatedly, assumptions -> conclusions -> conclusions -> ... -> conclusions, and every step on the way you are concluding in accordance with the laws of logic, you can be 100% sure that your conclusions are as true as your assumptions are. So, if you believe in your assumptions, you have to (and don't have a choice) believe in your conclusions. This is the power of logic and no other system of reasoning has it. Now, your assumptions can also be FALSE. If they really are false, then your conclusions can be ANYTHING (true or false) if you follow the laws of logic. This fact is a bit surprising but it follows from the definition of the implication in logic. The implication "p -> q" is false only when p is true and q is false. All other combinations of p and q make it TRUE. This has very profound consequences.
General proof books.... 1 How to prove it by Velleman 2 The book of proof by Hammack For analysis 1 Understanding analysis by Abbott 2 How to think about analysis by Alcock. Good luck
@@renatoteixeira3436 6th edition, can’t get any newer than that. I am going to use it along with the solution manual online this summer. I’ll be ready for real analysis in no time. 😁
There's no way to prove a theory. A theory can only be falsified, never proven. What you are asking for is a proof of a statement in some theory. That's a completely different thing. And to prove this (if it's really true), you should check out books on Graph Theory.
So what is the difference between doing a bunch of arbitrary-chosen math that just so happens to provide what you're looking for, and doing proofs? Plain language? Logic symbols? I legitimately don't see any difference, and I feel like mathematical proofs are unnecessarily restrictive to a predicate logic that doesn't really allow you to fully explain what's going on, outside of essentially writing out math in plain language with some predicate logic symbols.
Hello sir! This video is very useful but, I do have a doubt... Can't we prove a theorem without using the contradiction? Also , can a theorem have more than one proof? Please consider my questions . ~love from India
There is also a 5th technique of proving p->q that computer scientists love, but mathematicians hate, which is giving a list of every p and showing q is true for it. It's not so good if there is an infinite number of examples though...
Getting bogged down by linear algebra theorem & proofs. I find Linear Algebra by Kenneth Hoffman and Ray Kunze to be too much to take as many proof are left for the readers, nothing personal with that book, but it is provided as one reference book in Master degree and seems to be very standard one and i did not find another. Can you please provide some good references, course, books, videos for Linear Algebra theorems and proofs.
@@DrTrefor Thank you very much for making this video. it really helps a lot of people like me to not be afraid of attempting mathematical proofs and theorem. Thank you for kindly taking time on a personal request and sharing a book which might be helpful.
This video is why the internet was invented
This video was straight up perfect, don't understand why four people would dislike it
“Omg math smh” -people that click on math videos but don’t like it
Teachers that actually applies logic to their courses (including explanation of logic) are very clear…
Professor introduced this concept for 3 lectures, and this guy cleared up all of my confusion in 15 minutes. Perfect video
I had you at U of T for MAT137! I had a lot on my plate at the time so I didnt do very well, but Im back at school now and my Major is Mathematics because of your course. You're the most memorable prof I've ever had. Still coming back for your lessons. Thank you very much!
Woah! I have a lot of mixed feelings about that class, if you didn't know the coordinator for 137 that year Alfonso tragically passed away. But regardless nice to see you again and congrats on finding your way back to math!
Scaffolding Thinking! Clarity! A 68 yr old woman, not a mathematics major, minor, medium - 'your name' caught my attention 'once again' after two years of whooped-dog-feeling avoidance (due to so much rewiring occurring) of completing an online mathematical thinking course. Spent the morning rewinding your video between numerous other brief interruptions. Invaluable! Keep 'em coming. Returning to Art, persisting at Spanish, and squeezing forward a row to reclaim that seat in math class, yet... this Rural Elder Thrives! Number Theory seems to entice, too. Thank You. PS I recognize the shifts in temple hair (not beard) color, all too well! Chuckle.
You . Deserve . My . Tuition . Fees . More
Thank you for this. I think you make the same point that I came up with after failing my 1st Linear Algebra test. My Calculus 3 course is all calculations and execution of Algorithms which I'm doing fine in and didn't realize I was so used to them that I was not as familiar with the theorems or proofs. My linear algebra course blind sided me because the professor went heavy on the test with theorems for proving if something is always, sometimes, or never true when backing up your statement. I will apply your process for every definition I see to make sure I have a firm grasp of the definition and theorems
6:28 The "write down the definitions" method works *really* well IME for set theory and topology theorems! Like: "Prove that if A \int B = A then A \subset B" - writing A is a subset of B as "if a is in A, then a is in B" and write A intersection B as the set if elements x such that x is in A and x is in B, the theorem falls out immediately!
Thanks for this video!
It really made a lot of concepts clearer for me. Proofs have this fascinating property that sometimes they seem really arbitrary for me and I have no idea how someone can come up with that, but it’s nice to know that there is some sort of scaffold that you can use to rise to the conclusion!
pov: it's 24 hours before your exam and you landed in the right spot
great video and explanation
haha, good luck!
How did it go?
@@Basiljsk I passed, and I have this video among others to thank.
I am in your position right now
I have my final exams next week and i am so worried my whole geometry exam is based on proofs like for eg sss sas asa and converse of isos triangle theorem, isos triangle theorem and way moreeee
Long story short: the information in this video is great.^^
When I went through this, I realized that I knew almost everything that Dr. Trefor is talking about. This video puts together all information nicely, that I felt the "dots are connected" :)
I had to take a discrete math class once, and noticed a common issue that trips me up is remembering the definitions of things. For example in the case of "If X is even then X squared is even." I forgot what the actual definition of an even number was lol.
Professor I am a high school student from india preparing for my engineering entrance exam and want to understand maths to it's core and so I often search out mathematical informative videos on the internet apart from studying the regular course and this is the best video I found over the domain I worked upon, it just made me think to depth, it was a really simple video with a lot of knowledge being given
Thanks a lot
Thank you so much sir!!! Your videos are awesome!! Amazing teacher!
Theee minutes in and this is already so very helpful!! Thank you so much
Wow...very encouraging for me! I'm in the first steps of my "proof journey"!
This helped so much! I can’t thank you enough!
I'm so glad!
Thank you for post this. Blow my mind. You have a great channel. Greetings.
This sort of video is an excellent public service. Thanks!
The help of such videos is immeasurable
Now I think I get what proofs are 💯 thank you 👍👌
I'm from India .where we have to cover huge sllybus may be in just 2 months of one semster and my tution teacher also give me practical eg in maths ie you give about wife and dishes . But these eg are really helpful to learn maths
And then there was light 💡
Thanks for helping me understand this.
Very nice video. Very helpful. Thank you. 👍👌
This gave me better insight thank you so much!
Also I think it’s worth noting to write down the exact definition, e.g. something says 2 curves are tangent, define that as having a line that is the common tangent to each curve, rather than saying the two curves touch one another
Thanks! great explanation!
This video is great.
Thank you Sir
You are blessed man for this video
Thanks for this amazing video.
what books do you recommend for learning proofs?
You explained the logic of lagic, professor.Thank you.
Thank you so much!
Great video Dr Bazett. Are there any further readings you could recommend to dig into the weeds a little further?
When I taught myself Calculus I used to come up with my own derivatives and integrals, meaning I would throw in random function and try to diff/int them. So my question to you is can I apply this in terms of thinking up a random proposition and then try to show if it holds or not? I know I'll have to be precise with definitions and my logical steps but pretending were in fairyland where the rules are loose, is this possible? I'm asking this because if I run out of exercises I want to see if I can state some proposition and see where it takes me
Wow, ur too good at this
That helped me thanks 🙏
Hi Professor,
Do you use CAS software like Maple or Mathematica to help you understand or write proofs? If yes, would you mind doing a video about it?
Thank you!
You are a life saver!!
Super good 👏👏👏
Hello Prof,
Thank you for this video. This is more like a light in the tunnel.
However, this whole real analysis stuff looks totally strange and kind of challenging to me. Possible textbooks or links to solidified my understanding will be appreciated 👍.
Thank you Professor
Thank you sir
100k subscribers BAZZA!!!!!! yesterday I saw you were on 99.9K, I come today and BOOM! smashed it! GG
Haha thanks!!!
God bless you sir. God bless you so much.
this actually motivated me to think I can learn this
This was very helpful for me❤
Have my exam in 2 days, your saving my life :)
thanks so much
Tnx for the video, interesting!
AWESOME you may not know but this was like 100% gggreat for me. OMG SUPER-NICE CHANNEL.
Thank you for this video. I double majored undergraduate in physics and EE, the physics was focused on particles, and I’ve always felt weak in proof. The idea of playing around with a proof is revelatory, and some areas of proof come easier-perhaps that’s tied to my understanding of certain areas. I’ve had a block with proof in probability theory, but not with topology-perhaps it’s because I can visualize (referencing back to particles and fields) an analogy to “see” the math problem referenced, better. This video has expanded my understanding thank you. Is there a book you’d recommend for reading relative to mathematical proof?
I've found "How to Prove It: A Structured Approach", by Daniel Velleman to be quite good. It goes into detail on quantifiers, set theory, and proof techniques (induction, contrapositive, etc.)
@@jakedelyster3360 thank you-I’ll look it up!
so what I am understanding is that which proof method is best in which situation is just trial and error of trying each one see if it leads anywere if not try using the next method to proov is that correct?
I love this video and it really helped in my proofs, but ive found this question stating prove that x^3 -x is a multiple of 6 I know the answer now but how should I've tackled this question
I'm watching you from Algeria 🇩🇿
How do we use different proofs to prove the negation of cubic equation and its given negations of its x values? All the examples shows for odd and even intergers only.
Nice video
Thanks for doing this video. I am confused as to how you defined even numbers in your assumption as x=2p, where p is an element of Integers.
Aren't even numbers supposed to be numbers that are devisable by 2 with zero remainder?
And also, it is again weird for me that the conclusion is X squared = 2q, where q is an element of integers.
Theorems are such a weird thing that make no sense to me regardless of how long I stare at them and try to figure them out. Extremely frustrating to be honest.
Edit:
Actually it's making sense to me now haha :D , the integer definition part I mean.
Trefor I'm an engineering student i don't know what a proof is
I think it goes something like this. if pi ~= 3 then pi^2 = 9. g ~= 9, there for pi^2 = g.
@@DrTrefor Jokes aside, your channel is great and always helps me to clarify any doubts that I may have. Greetings from Argentina and keep up the great work!
Thanks aLot Prof
In the 'black box' that comes up at about 2:07 it seems that there is an error. The AND statement shows And: p or q. Shouldn't it be p and q ?
it should be p and q
Hey! Helpful video! Any tips on Actuarial Science?
@@DrTrefor Awesome tysm. I'm just in my first year of university. Will keep that in mind.
Professor, thank you so much for posting this video. I am trying to learn Proofs on my own, and the breakdown of how to study and do proofs is very definitely interesting and has practical value as well, esp the tip suggesting to come up with Geometrical pictures and the one where you ask to come up with concrete example that show that the proof holds. So there is both theoretical and practical value in what you are asking us to do. Will definitely take a few proofs and apply your steps to it. Thank you very much once again for a quality video.
Dr. Trefor Bazett is the real ambassador of spreading Mathematics in very easy manner.
He makes it look easy.
thx
NOTE: For the list of different logical structures, it says "And: p or q". Should say "And: p and q"
I'm not sure if this has been stated clearly in the vid, but the "do your manipulations" phrase means: proceed according to the laws of logic. The laws of logic have this wonderful property that they always lead from a true assumption to a true conclusion. ALWAYS. The laws of logic are called tautologies. If you use these to draw conclusions repeatedly, assumptions -> conclusions -> conclusions -> ... -> conclusions, and every step on the way you are concluding in accordance with the laws of logic, you can be 100% sure that your conclusions are as true as your assumptions are. So, if you believe in your assumptions, you have to (and don't have a choice) believe in your conclusions. This is the power of logic and no other system of reasoning has it. Now, your assumptions can also be FALSE. If they really are false, then your conclusions can be ANYTHING (true or false) if you follow the laws of logic. This fact is a bit surprising but it follows from the definition of the implication in logic. The implication "p -> q" is false only when p is true and q is false. All other combinations of p and q make it TRUE. This has very profound consequences.
Also proff by induction
8:25 was that a part that you missed to cut out? "so now we have the manipulations" part
hahah oops ya that can happen:D
@@DrTrefor Love how you reset your composure and restart your explanation. Also thank you very much for these great videos.
Sir, May I write turkish subtitle for this video?
Can you suggest any books on how to write math proofs and survive real analysis?
General proof books.... 1 How to prove it by Velleman 2 The book of proof by Hammack For analysis 1 Understanding analysis by Abbott 2 How to think about analysis by Alcock. Good luck
@@renatoteixeira3436 Thanks! I am waiting for Amazon to deliver the Velleman book. Should be here by Monday.
@@ntvonline9480 Hope you are getting the newer version since it also contains a section on number theory.
@@renatoteixeira3436 6th edition, can’t get any newer than that. I am going to use it along with the solution manual online this summer. I’ll be ready for real analysis in no time. 😁
Let G be a connected self-complementary graph. Then diam(G)=2 or 3.
How to proof this theory??
There's no way to prove a theory. A theory can only be falsified, never proven. What you are asking for is a proof of a statement in some theory. That's a completely different thing. And to prove this (if it's really true), you should check out books on Graph Theory.
Wow
So what is the difference between doing a bunch of arbitrary-chosen math that just so happens to provide what you're looking for, and doing proofs? Plain language? Logic symbols? I legitimately don't see any difference, and I feel like mathematical proofs are unnecessarily restrictive to a predicate logic that doesn't really allow you to fully explain what's going on, outside of essentially writing out math in plain language with some predicate logic symbols.
Watching this moments before my test
hope it goes well!
Hello sir!
This video is very useful but, I do have a doubt...
Can't we prove a theorem without using the contradiction?
Also , can a theorem have more than one proof?
Please consider my questions .
~love from India
Yes a theorem can have more than 1 proof. The Pythagorean theorem has more than 370 proves.!
Yes you Can proof in any way you are good in and it is correct
Here a more challenging question you should ask yourself if things are getting boring: Is the number of proofs of a theorem finite?
There is also a 5th technique of proving p->q that computer scientists love, but mathematicians hate, which is giving a list of every p and showing q is true for it. It's not so good if there is an infinite number of examples though...
You are a god, I spent 4 years as a pure math major and I don't think I understood how proofs work exactly. Now I do.
I have one week before my test. Live and breathe proofs for the next 8 days
Good luck!!!
Getting bogged down by linear algebra theorem & proofs. I find Linear Algebra by Kenneth Hoffman and Ray Kunze to be too much to take as many proof are left for the readers, nothing personal with that book, but it is provided as one reference book in Master degree and seems to be very standard one and i did not find another. Can you please provide some good references, course, books, videos for Linear Algebra theorems and proofs.
@@DrTrefor Thank you very much for making this video. it really helps a lot of people like me to not be afraid of attempting mathematical proofs and theorem. Thank you for kindly taking time on a personal request and sharing a book which might be helpful.
May I contact on email with you sir
I got more confused. thanks.
I have discovered one theorem
Incomplete example , you cannot use unproved argument. Sqr(x) := Sqr(p) , there are equivalent statements. U cannot define q as sqr(p)