Four Basic Proof Techniques Used in Mathematics
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- เผยแพร่เมื่อ 4 ก.ค. 2024
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Part 1: • Introduction to Fundam...
Extra Proof by Contradiction with some death intrigue (huh?!) • Murdered Over a Math P...
New to proving mathematical statements and theorem? I this video I prove the statement 'the sum of two consecutive numbers is odd' using direct proof, proof by contradiction, proof by induction and proof by contrapositive.
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
It's always weird how tutorials on TH-cam do such an amazing job at describing specific topics in less time than a professor. I'm lightweight pissed off that I pay so much in tuition only to go online and learn the same exact material for free..
Nathan Davidson I’ve also happen to come to that conclusion. I believe it might be due to lack of distraction. I personally find it more comfortable taking my time at home than in school.
@@officialsterlingarcher In college professors are limited by time so they really have to meet or exceed their lessons for the next class. In the end, they have to rush through lessons because some topic may be particularly hard.
@@LocaColaDavid mèo
Nad
2:37 Direct Proof
5:45 Proof by Contradiction
10:30 Proof by Induction
17:30 Proof by Contrapositive
Great video explaining the proofs, horrible choice of example as it doesn’t cover anything remotely important.
Man, this is my first and will probably be the only TH-cam 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.
happy to help my fellow peeps out there
@@patrickjmt peep
10:!8 PM
10/20/2022
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Before you request his presence in the pantheon, understand he makes big bucks from ad revenue.
So all the mushy sentiments and love that is gushing are directed at someone whose goal is to profit immensely from their expertise AND there is nothing wrong with that of course.
So many apparent acts of altruism are similarly motivated.
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
every time school makes me feeling like dying, I come and watch your videos and my will to live is restored. bless you man
Damn you read my mind, I got a discrete math test with this on Friday!!!!
Omg I really needed this video, perfect timing, this video helped me sooo much, thank you sir 🙏🏾
Oh man I had three people come up to me for help with proofs, glad this got uploaded!
I literally learned this today. Thank you!!
I love your videos. Great job!!
Hello Mr. Patrick Sir, how would you like your name spelled on my college degree ?
LMFAO!
HAHAHAHA
Wow.👏🏼
Yo,he is life saver
Loooool
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.
Thank you! I had a test coming up this Friday and I hope you will put out more examples.
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.
@@Calculus99 Who are you to tell him that? Are you his teacher? I personally wasn't told of definitions in our course too.
@@Calculus99 You spew nonsense.
@@Calculus99 wow you need to back off :/
@Adam Taurus Thank you.
@@Calculus99 instead of pretending you were attending everyone's lecture, go back to playing your fantasy soccer games and pretending you were on the pitch.
I am so surprised you dont have more views. Discrete has been killing me and this really connects some of the dots. Thank you.
You explain better than my math prof. Thanks for making this video
I'm about to start grad school for Computer Science and have decided to brush up on my fundamentals. This video is super helpful!
good luck!
wow man, you ROCK! what an awesome teacher you are
Right to the point. Excellent!!
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
fame and money, hahahahahahaahhaahahahahahahahahahahahah
and thanks for the kind words :) i still make videos for the same reason as when i started; no reason to change them! i guess i could make the channel more personality driven but i don't want that
😂😂
Seriously thank you 😊I never knew such things existed!
This was a good idea. Thanks
You are a GOOD teacher !
Amazing video! Thanks so much :)
Thank you so much ❤ . Buddy. It is just mind blowing🤯
I can not thank you enough buddy for your help.Thanks for your service!
you are welcome, dill pickle
Thanks for the vid, it was informative indeed.
Appreciate it.
you saved my life!!! thank you sir
Great job!
Nice. Thanks a lot.!
great lecture note
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.
I always felt some kind of excitement coming from someone who was about to proof something. Prove me wrong
Assume that a person is excited.
But suddenly the ink of his/her pen dies.
Therefore, the person can't be exited.
It contradicts.......
Great video thanks!
Thank You So Much! Was totally lost before I saw this video!
Nice name
thanks so much. this is very elaborate
Could you also count “proof by exhaustion” as a proof?
I prefer "proof by confusion"
you obviously know the answer if you're asking that question
@@wachowski9525 yeah I don’t remember why I asked this question. Weirdly I do remember asking this question and watching this video
@@theseafaringsaxophone440 I prefer proof by induction
@@pjgcommunity3557 Proof by induction is more elegant and the most reliable proof method I've seen
Really good video thanks a lot :)
Had my first class of aylasis for enginers but i was late,this helped a lot thanks man
Thank you !!!
Just in time for linear algebra proofs!
had to look it up in english since spanish wasn't cutting it, thank you so much
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
you are amazing, love from PH!!!
Sir your video very helpful thank
very good
thanks so much
Thanks
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.
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!
They teach algebra in college??
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 :).
8th grade?
@@loofus9133 Well, it depends on the countries and levels but the course I took is college level.
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
PatrickJMT the OG youtube tutor!
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
patrick got me thru high school calc, hope youre doing great man!
i'm doing pretty well, thanks :)
@@patrickjmt proof by induction is used fro natural nos. Right?
I love the methods of proof.
Awesome!
Watching this in 2020, man you are great thanks very much you helped me pass my test.
Same (watching in 2020) :)
Glad I could help!
@@patrickjmt need more for contrapositive
Thank u souch sir
thanks
Awesome videos! Wish I found you earlier.
Wonderful
I wish my professor teach like you
where can i get more example videos? i really like the way you teach
well, another math class another semester of PatrickJMT, here we go...
please turn your book to page....
Hello. So I wonder which are advanced techniques? Could you please share books for basic and advanced techniques? Nice video 👍
You should put time stamps on your longer videos
Would you be able to do videos on automata? Specifically drawing state diagrams for languages that only accept some strings of 'a' and 'b'
Plz add a lecture on rectangular corrdinates 3 d sketching
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.
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?
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.
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
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?
you is grate
Sir can u plz put a video on teaching how to find the rank og a matrix
That is basic and useful
Who else got lucky taking Discrete this semester? haha
Me 😭
mee, how did you do in your class?? This class is really intimidating compared to calculus 2
Discrete math was easier for me than calculus 2 was. But there were people in my class that were struggling, it’s different for everyone
😥
You saved me throughout discrete math, so grateful. Keep it up
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 ???
that sharpie scratch.....such ear torture. but great explanation
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?
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.
Thanks, but I hate the dry sound of sharpie on paper. Its like the sound of chalk on chalkboard.
So what do you have against chalk and chalkboards
Can we write exception in proof? If we can't, how do we explain in another way?
Congratulations.
Notification: I would like to inform you that I plan to include this video among the videos reviewed in my article "Analysis of TH-camTM Videos and Video Comments on Mathematical Proof Methods". Kind regards.
Please can you solve more examples for all the proofs you mentioned. I still don't understand
where did you get the definitions? Are you supposed to make them or will they be given to you
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.
Hythloday71 ur not proving anything, you solving for x when y = 0
I think that could be considered direct? I tend to agree with the above comment, but maybe the statement being proven is "prove that if this quadratic equation has roots this formula will work". Showing how you derive it does seem like a direct proof since you work with the information given, apply true things, to arrive at the desired conclusion?
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
This is really sinking.. I almost floated in any lecture but I now understand.
without looking at other proof methods how would i know that 2k+1 is another way to represent and odd number
I have a question abut the contraposition proof... If you negate the antecedent "a and b", would this not result in "a or b"?
~(A ∧ B) = ~A ∨ ~B
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).
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.
9:52 I've seen some very elegant proofs by contradiction though. Such as the proof by contradiction of the fundamental theorem of arithmetic :-)
Also proof by counter example
well, that is usually used to disprove something
that last one was a bit confusing. How does K+1 being the 'successor of k' imply a and b cannot be consecutive?
Yeah that got me confused as well. 2K+1 is ODD.
but the assumption is: that A + B is NOT ODD. But it is ODD?
Guys, the assumption is that a+b is NOT odd which means they CAN NOT be written in the form 2k+1(where k is an arbitrary integer). But since 2k+1 is equal to k+(k+1) this means that it can NOT also be written in the form k+(k+1). Hence a and be can not be consecutive.
So once again:
a+b can NOT be written in the form 2k+1. But 2k+1=k+(k+1). So can NOT also be written in the form k+(k+1). So if it can NOT be written in the form k+(k+1) then that means a and b can NOT be consecutive
So it's much simpler than u think.
I was also confused on the reasoning behind the Contra-Positive approach. I think if you assume a+b NOT to be odd then you can say it is even or a + b = 2k. Afterwards, you express, 2k = k + k which are NOT two consecutive integers and you attempt to show that they cannot be; (k - 1) + (k + 1), (k - 2) + (k + 2), etc. As you go on, the numbers drift further and further away. Since by definition, consecutive integers are k and k + 1 OR k and k - 1, we see that a, b CANNOT be consecutive.
If i am given a question to prove. How can i decide that i have to use one particular proof strategy?
If a question says prove that blah blah blah holds true for all integers try to use induction. If a question says prove that such and such a statement is true or false, but the question has nothing to do with integers then you should try one of the oo the other methods.
If a problem says show that a rule has these constraints assume those constraints don't hold then use proof by contradiction to demonstrate that those constraints would have to hold.
Those are a few ideas, but there is no general rule. Proving things is not an exact science. Crafting the perfect argument (which is what a proof is) often requires trial and error. These are just places to start. Some problems are so difficult that there might not be an obvious way to use any of these methods to solve a given method. My advice is use trial and error. If that doesn't work keep trying these methods, but look for different angles to attack a problem from. If all else fails think outside the box.
4:50 how is b = 2a? what if a = 80? b =/= 161. can you help me out on tht?
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?