Man I just want to let you know I started using this channel back in 2012 when I started calculus. The funny thing is, for some reason I had some weird feeling like I was cheating by using your channel to help me understand the material. It really makes you think, I was so used to crumby teachers that when I actually learned something it felt like using cheat codes in a video game. I just wanted to let you know you are really doing a service that is priceless, there is no way I would have passed my math courses without you. Thank you

2:37 Direct Proof 5:45 Proof by Contradiction 10:30 Proof by Induction 17:30 Proof by Contrapositive

Man, this is my first and will probably be the only KZfaq comment I make; you're a god among men. I owe it to you for making it a lot easier to understand a huge chunk of the material I study at college. What you're doing is priceless and I want to thank you from the bottom of my heart. You're an amazing person for giving this priceless service out for free.

every time school makes me feeling like dying, I come and watch your videos and my will to live is restored. bless you man

Dear Patrick , Hopefully, you have been faring well. I just wanted to thank you for the effort that you have been devoting to clarify things through your consecutive episodes. I'm a PhD student at EE department and still get back to your videos to learn. It is been almost 7 years or so. Thank you again. Ibrahim from KSA

Omg I really needed this video, perfect timing, this video helped me sooo much, thank you sir 🙏🏾

1 minute and 20 seconds in already you have shown more than the book and the professor teaching our course. It was never explained that there were definitions for proofs.

Oh man I had three people come up to me for help with proofs, glad this got uploaded!

Damn you read my mind, I got a discrete math test with this on Friday!!!!

Thank you! I had a test coming up this Friday and I hope you will put out more examples.

Hello Mr. Patrick Sir, how would you like your name spelled on my college degree ?

Patrick I'm just now stumbling upon your videos in my Freshman year of college and I think it's safe to say I love you.

I literally learned this today. Thank you!!

All these years have passed, that you have fame and money now, but you're still an awesome teacher and most importantly you did not change the way you make these videos. Thank you

I am so surprised you dont have more views. Discrete has been killing me and this really connects some of the dots. Thank you.

I love your videos. Great job!!

I'm about to start grad school for Computer Science and have decided to brush up on my fundamentals. This video is super helpful!

You explain better than my math prof. Thanks for making this video

Seriously thank you 😊I never knew such things existed!

Thanks for the vid, it was informative indeed. Appreciate it.

I always felt some kind of excitement coming from someone who was about to proof something. Prove me wrong

Could you also count “proof by exhaustion” as a proof?

Right to the point. Excellent!!

This was a good idea. Thanks

wow man, you ROCK! what an awesome teacher you are

i never learned to really proof anything with defining things because i learned how to do calculus only and understand, this is so useful. smh how did i pass with an A and now knowing this tysm man

Thank you so much ❤ . Buddy. It is just mind blowing🤯

I can not thank you enough buddy for your help.Thanks for your service!

Amazing video! Thanks so much :)

Had my first class of aylasis for enginers but i was late,this helped a lot thanks man

Just in time for linear algebra proofs!

You are a GOOD teacher !

thanks so much. this is very elaborate

great lecture note

you saved my life!!! thank you sir

Really good video thanks a lot :)

had to look it up in english since spanish wasn't cutting it, thank you so much

Great video thanks!

Nice. Thanks a lot.!

Great job!

I just wanted to say thank you for your videos. I was in an Intermediate Collage Algebra course, but discovered through doing the homework and using your videos to help with that homework, that this was the wrong class for me to be in. It moved so fast and its assumed we already know things like how to find the slope/y intercept etc....so no only did I finally have a real question for once, I was also able to save myself several more very long months of being lost and stressed out. Usually, I would have just tried to stick it out, instead, met with my counselor and discovered that I don’t even need that class for my major since my transfer school has changed!! So long story short, thanks man!! Your videos are the best and easiest to follow along with and I will most certainly be referring to you more next semester when I take the correct Algebra course!

Great video! I just want to point out that the proof by induction really only holds for the positive integers, since the base case is n=1, and you proved p(n)-> p(n+1) but you didn't prove p(n+1)->p(n). I believe a second half of the proof is needed to cover integers less than 1.

Sir your video very helpful thank

Thank You So Much! Was totally lost before I saw this video!

Thank you !!!

patrick you are an amazing tutor..but what about this definition: an integer a is a perfect square if there is an integer b such that a=b^2..thank you

You're really amazing you don't know how many times you saved my life, so could you save me once again by please please do a video about bessel functions and also green function i would be so grateful

Omg, I didn't know you have a video about proofs. Oh my, I'm still struggling with that class 😢😢. But your other courses helped me to pass my class :).

very good thanks so much

I was taking MAT243 at Arizona State and.. man the proofs kicked my butt. I dropped the class. I'll be retaking the class in the spring of 2020 and definitely will be recalling these videos on proofs.

you are amazing, love from PH!!!

It took me a while to get my head around the proof by contradiction. I can understand why it's not really in favor with mathematician.

Hello. So I wonder which are advanced techniques? Could you please share books for basic and advanced techniques? Nice video 👍

where can i get more example videos? i really like the way you teach

Thank you Patrick you are the best when I don't understand something in class I don't mind because I know I've got you...great videos I really wish you were my lecturer

Would you have to say "immediate" successor in the proposition or have you not done that deliberately? The rule with addition of odd and even means that if a + b is odd, one of a and b must be odd and one must be even. So you don't need to?

PatrickJMT the OG youtube tutor!

Would you be able to do videos on automata? Specifically drawing state diagrams for languages that only accept some strings of 'a' and 'b'

patrick got me thru high school calc, hope youre doing great man!

Thanks

hello and good morning.. i have a question like this, Let A: {n element of Z | (there exist k element of Z) [n = 4k+1]} and B: {n element of Z | n is odd}.. How do I prove that A is a subset of B and disapprove using counterexample that B is not a subset of A?

I love the methods of proof.

Watching this in 2020, man you are great thanks very much you helped me pass my test.

0 divided by any # is equal to 0. Any # divided by itself is equal to 1. The first statement would lead us to believe 0/0 = 0. The second would lead us to believe 0/0 = 1, there must therefore definitionally be something wrong with these axioms; as it is universally accepted that no # can be divided by zero. If we adjust the first axiom to, 0 divided by any # except 0 is equal to 0. We can resolve the contradiction. 0/0 = 1 (this proof will remove "holes" from all rational equations) Further, if we observe rational equations; the asymptote (where the denominator = 0) could be considered to be both + and - infinity. As positive infinity is approached to one side of it, and negative infinity is approached on the other side of it. In order for the asymptotes to be consistent with the rest of the graph, any # divided by zero must be equivalent to + and - infinity.

Awesome!

well, another math class another semester of PatrickJMT, here we go...

You should put time stamps on your longer videos

Thank u souch sir

Awesome videos! Wish I found you earlier.

Can we go from q to p ? can we say that if c is odd, then there are two consecutive numbers such that a+b=c ???

Sir can u plz put a video on teaching how to find the rank og a matrix

Plz add a lecture on rectangular corrdinates 3 d sketching

Can we write exception in proof? If we can't, how do we explain in another way?

Wonderful

thanks

Thanks for these videos. Do you by chance have any general logic courses not necessarily in math but in a generalized case where you talk about inductive and deductive reasoning and other logical reasoning methods including coverage of all mistakes in logic (fallacies).

I wish my professor teach like you

Got some idea about how to do the proofs hello brother can u tell me how to remember the proofs when there are many theorems in pg courses

When you prove the formula for quadratic roots, what kind of proof is that ? I like to call it proof by algebraic generalisation. It doesn't quite fit the 'direct proofs' in most 'abstract algebra' propositions and theorems.

I'm still confused by proof by induction. What if you had a statement all positive integers are less than 10? Intuitively we know it's false, but it works for k between 1 and 8. If you stop at k+1 you wouldn't know that. I come from a programming background so a lot of ideas in math are close but not quite familiar.

Please can you solve more examples for all the proofs you mentioned. I still don't understand

Who else got lucky taking Discrete this semester? haha

I have a question abut the contraposition proof... If you negate the antecedent "a and b", would this not result in "a or b"?

Thanks, but I hate the dry sound of sharpie on paper. Its like the sound of chalk on chalkboard.

Is this a valid proof as well ? Consecutive numbers alter between odd/even, so I rephrased consecutives = odd as odd+even=odd Given # (2n) = (even) # (2n+1) = (odd) # { k, n | Z } Assume # (2n) + (2n + 1) = (2k + 1) 2(n + n) + 1 = 2k + 1 2(2n) + 1 = 2k + 1 2(2n) = 2k 2n = k (2n) + (2n + 1) = (2k + 1) 2n + 2n + 1 = 2(2n) + 1 2n + 2n + 1 = 2n + 2n + 1 # Done Since the equation shows true, and the steps are true, the assumption holds true. Right?

without looking at other proof methods how would i know that 2k+1 is another way to represent and odd number

That is basic and useful

you is grate

Congratulations. Notification: I would like to inform you that I plan to include this video among the videos reviewed in my article "Analysis of KZfaqTM Videos and Video Comments on Mathematical Proof Methods". Kind regards.

If i am given a question to prove. How can i decide that i have to use one particular proof strategy?

This is really sinking.. I almost floated in any lecture but I now understand.

that sharpie scratch.....such ear torture. but great explanation

Good Video, but I think the induction could be clearer. Usually you show that the x+1 case splits into the x case and some other part. The x part is already prooven, the other part doesn‘t contradict it, so it is true for x+1. F(x+1)=x+1+x+2 = x+x+1+2 = F(x) +2. We know F(x) is true for some x, so F(x+1) is F(x) + 2 which must be shown is also odd.

where did you get the definitions? Are you supposed to make them or will they be given to you

4:50 how is b = 2a? what if a = 80? b =/= 161. can you help me out on tht?

that last one was a bit confusing. How does K+1 being the 'successor of k' imply a and b cannot be consecutive?

9:52 I've seen some very elegant proofs by contradiction though. Such as the proof by contradiction of the fundamental theorem of arithmetic :-)

This proof is the only thing which I can not understand it in mathematics😂😂😂 I don’t know how l will do in tomorrow exam

What about proof by seduction?

I start discrete math in a week :,) someone please help me lol