I’ve been in two discrete math focused and algorithm analysis courses, I’ve been a TA for one of those courses as well, and not once have i ever seen such a clear explanation of why we do what we do for both weak and strong induction, and how to replicate this consistently and clearly. And you clearly care about your students understanding too!
ive spent the last two years of my undergrad watching countless videos of people teaching math, and this is the first time I've commented because you've got me smiling while learning about strong induction! Your enthusiasm and energy is contagious, and your teaching both clear and concise. Thank you for a great video!
Hi doctor. I just meet you recently. And After I watch your video I said Why did not I found it before :D Haha whatever I find you. I just have one rewiew about the lesson. your 2 number's written as a 'a' When I saw the question I thoght that was a 'a' when you start the solve the question Than I noticed that it was not a 'a' I do not know did the other people consider about that think but I just wanted write. And except for all of this the lecture was very clear and also it was clear to you love your job to do
Undergrad student here and I think if we had more teachers like you, people would understand everything in one attempt! Your enthusiasm and energy is infectious!
I have been searching for strong induction courses on TH-cam for a couple of days. No other course can explain this particular concept more clearly than this one.
I appreciate you 🙏🙏 in addition to this, please share Ethiopian University freshman courses .(general physics, mathematics, logic and critical thinking, anthropology and Introduction to emerging technology.
Usually I just watch videos and move on after grabbing things I want, but this was too helpful that I couldn't resist commenting! GREAT WORK!! Thank you so much for this! This is my first comment ever on any video , all credits to your brilliance
Thank you for the super video Dr. Forgive me but why is one allowed to assume all the steps from k(1) all the way to k(n) are true? This is the only part of strong induction I don't understand. I have scoured TH-cam and the internet for an explanation but cannot find one. Every source I have come across just states you can assume it but I haven't found an explanation as to why. I'm sure it's something obvious that is eluding me :(
So we have to be careful with our wording here. Technically we do not assume k(1) up to k(n). We want to prove an implication. that means: *IF* k(1) to k(n) are true *THEN* k(n+1) is true. now. you are correct as we prove the implication (inductive step) we start off suppose/assume k(1) to k(n) are true and show that with this assumption k(n+1) is true. this is how we prove an implication (due to logic/truth tables). Ok now to your question. One can prove that strong induction is equivalent to (regular) induction. These are both equivalent to well ordering principle of natural numbers. There are many references in textbooks showing the equivalence of different types of induction. hope this helps.
@@DrValerieHower thank you! How do we know when to use strong induction bs just regular induction? Should strong induction always be used when recursive sequences are involved??
I always tell students that you don't need to make a decision in advance. Just try working with the inductive step. If you realize you can prove for k+1 using only the statement holding for n=k. then (regular) induction is just fine. and then you have one 1 case to check in your basis step. However if you realize you need the statement to hold for some value(s) of n that are less than k to prove for k+1 then this will be strong induction. You will also need to determine how many cases to check in your basis step.
Thank you so much for this video!!! I'm currently preparing for math proof test, and the questions provided in the video were clear and intersting! Thank you!!
You are the type of professor I wish to be in my future 🥰 straight forward and as clear as possible! I have to work really hard but when I finally understand something I’m golden and because of this I think I’m better at explaining where people might have trouble which you do a great job of! I also appreciate you writing out these proofs! Most professors are lazy and short hand everything which can lead to confusion. Thank you so much Dr! You are amazing!
I think inductively, and I have stamps of 3-cent value and 5-cent value. If I want to make (k+1) cents in an inductive way, I can take postage made for (k-2) cents and then add one 3-cent stamp. This would give (k-2)+3=k+1 total cents. This is "3 steps back" and why I check 3 cases in my basis step. Does this help?
for some reason my professor didn't explain how the inductive step is different for mathematical and strong inductions, this made my life easier. Thank you!
I’ve been in two discrete math focused and algorithm analysis courses, I’ve been a TA for one of those courses as well, and not once have i ever seen such a clear explanation of why we do what we do for both weak and strong induction, and how to replicate this consistently and clearly. And you clearly care about your students understanding too!
Wow. Thank you so much for your kind words. I really appreciate the feedback.
ive spent the last two years of my undergrad watching countless videos of people teaching math, and this is the first time I've commented because you've got me smiling while learning about strong induction! Your enthusiasm and energy is contagious, and your teaching both clear and concise. Thank you for a great video!
You are welcome. Wow! Thank you so much :). I appreciate the feedback and kind words. Best wishes to you!
I love the enormous enthusiasm you put into the videos
You are welcome! I really appreciate the feedback. And thanks so much for the kind words.
Hi doctor. I just meet you recently. And After I watch your video I said Why did not I found it before :D Haha whatever I find you. I just have one rewiew about the lesson. your 2 number's written as a 'a' When I saw the question I thoght that was a 'a' when you start the solve the question Than I noticed that it was not a 'a' I do not know did the other people consider about that think but I just wanted write. And except for all of this the lecture was very clear and also it was clear to you love your job to do
Thank you for your comment. I appreciate the feedback. I do not believe I have errors in this video, but my apologies if something is unclear.
wish I had you as my prof which is passionate and on point rather than the current one who has 0 interest in what students are understanding.
Beautifully explained...
I have a test tomorrow morning and you saved me...
Thank you very much...
And keep smiling 😊😊😊
Thank you so much for your kind words. I appreciate the feedback!
Your smile ..feels me that how easy is math.... awesome lecture....
Love from India.❤️
Thank you!!
Undergrad student here and I think if we had more teachers like you, people would understand everything in one attempt! Your enthusiasm and energy is infectious!
Thank you so much for your kind words!! Best wishes in your studies :)
Love the energy and the clear explanations- thanks!
You are welcome! Thank you so much for your kind words. I appreciate the feedback :)
extremely well made!
Thank you so much.
I have been searching for strong induction courses on TH-cam for a couple of days. No other course can explain this particular concept more clearly than this one.
Wow thank you for the kind words! I really appreciate your comment.
I appreciate you 🙏🙏 in addition to this, please share Ethiopian University freshman courses .(general physics, mathematics, logic and critical thinking, anthropology and Introduction to emerging technology.
Usually I just watch videos and move on after grabbing things I want, but this was too helpful that I couldn't resist commenting! GREAT WORK!! Thank you so much for this! This is my first comment ever on any video , all credits to your brilliance
Wow! You are welcome. Thank you so much for the kind words :)
Thanks for ur support ❤
This was great! THANKS A LOT!! :)
You are welcome!! Thanks for the comment :)
What is the difference between strong and weak induction?
Thank you for the super video Dr.
Forgive me but why is one allowed to assume all the steps from k(1) all the way to k(n) are true? This is the only part of strong induction I don't understand. I have scoured TH-cam and the internet for an explanation but cannot find one. Every source I have come across just states you can assume it but I haven't found an explanation as to why.
I'm sure it's something obvious that is eluding me :(
Yeah I have the same question too
So we have to be careful with our wording here. Technically we do not assume k(1) up to k(n). We want to prove an implication. that means:
*IF* k(1) to k(n) are true
*THEN* k(n+1) is true.
now. you are correct as we prove the implication (inductive step) we start off suppose/assume k(1) to k(n) are true and show that with this assumption k(n+1) is true.
this is how we prove an implication (due to logic/truth tables).
Ok now to your question. One can prove that strong induction is equivalent to (regular) induction. These are both equivalent to well ordering principle of natural numbers. There are many references in textbooks showing the equivalence of different types of induction. hope this helps.
@@DrValerieHower thank you! How do we know when to use strong induction bs just regular induction? Should strong induction always be used when recursive sequences are involved??
I always tell students that you don't need to make a decision in advance. Just try working with the inductive step. If you realize you can prove for k+1 using only the statement holding for n=k. then (regular) induction is just fine. and then you have one 1 case to check in your basis step. However if you realize you need the statement to hold for some value(s) of n that are less than k to prove for k+1 then this will be strong induction. You will also need to determine how many cases to check in your basis step.
Was very confused with the stamps problem you blowed up all the contents at once
There are very few teachers who make us realize that mathematics can be fun and interesting. You are one of them. Keep up the good work.
Wow thank you!! I appreciate your kind words. :)
Great Lecture! Thank you
You are welcome! thanks so much for the feedback :)
Love the passion and honestly one of the best explanations I've come across among all other videos
Thank you so so much! I appreciate the feedback :)
Best Lecture on YT regarding Strong Induction
Wow thank you so so much!!
Thank you so much for this video!!! I'm currently preparing for math proof test, and the questions provided in the video were clear and intersting! Thank you!!
You are welcome!! i appreciate the feedback :)
You are the type of professor I wish to be in my future 🥰 straight forward and as clear as possible! I have to work really hard but when I finally understand something I’m golden and because of this I think I’m better at explaining where people might have trouble which you do a great job of! I also appreciate you writing out these proofs! Most professors are lazy and short hand everything which can lead to confusion. Thank you so much Dr! You are amazing!
You are welcome. Thanks for the comment and best wishes with your future goals. :)
Thank you for the content. It was really helpful and you explained it thoroughly.
You are welcome. I appreciate the feedback :)
Thank you so much, strong induction finally clicked for me!
You are welcome! this is so wonderful to hear. thanks for the comment :)
Superb teaching skills! Thank you
You are welcome! Thank you for the comment and kind words.
you the best
Thank you!
Amazing lecture, I always watch this video to review strong induction!
Thank you so much!!!!
One of the most helpful videos on Strong induction out on YT
Wow thank you so much!!
Love the way you teach. ❤
Thank you! I appreciate the feedback.
best video iˋve encounter, thanks!
Wow, thank you so much! And you are welcome :)
Hi Dr.Valerie. For the last problem, I didn't understand the reasoning about where k-2 came from.
I think inductively, and I have stamps of 3-cent value and 5-cent value. If I want to make (k+1) cents in an inductive way, I can take postage made for (k-2) cents and then add one 3-cent stamp. This would give (k-2)+3=k+1 total cents. This is "3 steps back" and why I check 3 cases in my basis step. Does this help?
@@DrValerieHower so more generally, what is the trick to find the (k-2) for inductive step in other problems?
Thank you Prof!
You are welcome!
You lost me at 3:15. Why did we choose 6 as the common denominator to become 6(K+1)^2/6?
You are the GOAT. Run for office, I'll vote for you
:) I appreciate the feedback. Thanks!
nice video ,topic simplify very esay👍👍👍👍
Thank you so much for your feedback.
I love you. Thank you for your effort!
You are welcome! Thanks for the comment :)
Loved your energy and the way of teaching!
Thank you so much! I appreciate your kind words.
Thank you Dr, u helped me a alot
You are welcome. I am glad this was helpful and appreciate your feedback.
What does S(n ) mean? Please explain.
Thank you 🙏
You are welcome! Thanks for the comment. :)
Amazing video, ty!
My pleasure! Thanks for the feedback.
17:55 Why is k >=2 when n>=3, doesn't that make 1
I'm not sure where you are getting that statement. I assume k>=2 in order to have k+1>=3. This is needed in the inductive step. Thank you.
Tysm !
You are welcome!
Is this not a weak induction?
Never mind, I didnt finish the video when I said this.
Correct I started with one more example of induction (weak) before moving into strong induction.
for some reason my professor didn't explain how the inductive step is different for mathematical and strong inductions, this made my life easier. Thank you!
You are welcome! I appreciate the feedback.
i was worried about my test . but it your smile which has given a motivation to study more thanks ..😃😃
Great! I am happy to hear. Thank you for the feedback!
your smile is sooo beautiful
Thanks so much for your kind words.
Hii mam, you are so beautiful.