Induction: Inequality Proofs
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- Опубліковано 30 вер 2024
- Proving inequalities with induction requires a good grasp of the 'flexible' nature of inequalities when compared to equations. Make sure that your logic is clear between lines!
For more mathematical induction proofs with inequalities, try these:
Inequality Proof Example 1, Σ(k = 1 to n) 1/k² ≤ 2 - 1/n:
• Induction Inequality P...
Inequality Proof Example 2, n² ≥ n:
• Induction Inequality P...
Inequality Proof Example 3, 5^n + 9 lesser than 6^n:
• Induction Inequality P...
Inequality Proof Example 4, n! greater than n²:
• Induction Inequality P...
Inequality Proof Example 5, 2^n ≥ n²
• Induction Inequality P...
Inequality Proof Example 6, [2^(2n)]*(n!)^2 ≥ (2n)!
• Induction Inequality P...
For videos on other kinds of mathematical induction, see my playlist on this topic: • Playlist
I'm Mr. Woo and my channel is all about learning - I love doing it, and I love helping others to do it too. I guess that's why I became a teacher! I hope you get something out of these videos - I upload almost every weekday, so subscribe to find out when there's something new!
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i like the way he teaches. it feels like he's just having a conversation rather than presenting
I didn't expect him to whistle at the beginning. What a surprise. It means he is *happy.* It must mean something good has happened on that day or week.
@@pinklady7184 psychology student
@@pinklady7184 I just think he enjoys his job.
I like his videos but the writing is pretty small, needs to be much larger.
He has an easy style of teaching. At home with himself, his students and the subject matter. I sub teach HS Algebra & Chemistry in thd Long Beach California and this stuff still stumps me even though I had the class 30+ yrs ago.
if x>=6k ,6k>=2k, wouldn't be that x>2k not x>=2k because if x=2k then it can't be >6k?
i love you.
you make my college professor look like an idiot, thanks for helping me pass discrete maths bro :,)
this
my professor makes me look like an idiot. He makes my professor look like an idiot. :)
Great vid, but I honestly don't think you look like a teacher, i thought you were a brillant student dressed like a teacher, thanks btw!
Thank you Eddie for your teachings, your students are truly blessed
I've never seen a Asian with an Australian accent before. He's hot and smart. Great video.
You've never heard that you're hot and smart before? lol and you read the comments?! I guess it is your channel. I hope that didn't come off as disrespectful (about the accent/ethnicity thing). Wasn't my intent at all. I learned more from this one video about induction than half a semester on induction. I'll be scanning your videos for help on epsilon-delta proofs too! Thank you so much, Professor/Dr. Woo!
***** Haha thank you Mr. Woo! And check your direct youtube messages if you're bored and get the chance! =)
***** One teacher to another. You are doing a good job.
I agree, great job. The book didn't explain it nearly as well as you did.
Looked at a few books, videos and web pages, but only after watching your video, induction is finally beginning to make sense to me. Thanks and keep up the great work.
All your induction videos are great :)
I didn't do Higher Maths at school so am now doing an equivalent at university this semester so I can pick up maths next semester. We did a single lecture on induction and a few examples and I was really struggling, but I've really started to get it all clear in my head now. Thanks :)
Thanks A LOT for this! You're a great teacher I finally understood
(k + 1) k! > (k + 1) 2^k
(k + 1) > (k + 1) 2^k what happened to k! on this line at 12:30
thies2us (k+1)! = (k+1)(k)! = (k+1)(k)(k-1)! and so on
muy buena aunque no entiendo el idioma pero el ejercicio lo entendi todo saludos desde peru
This is so much more intuitive than the way I was taught. Awesome video.
Thanks a lot, I found it very helpful.
The second inequality is actually also true for n=4
4!=4*3*2*1=24, 2^4=16, 24>16
I LIKE THIS TEACHER PLZ CAN YOU COME TO OUR SCHOOL
This is what teaching should be like, this is professional teaching, I'm taking a discrete math class in college and its making me miserable, these PHD professors are useless!!!!! How come a UA-cam tutor is 100000% better than those useless PHD professors at my university, they just over complicate simple things, this is so refreshing. Thank You!
I fucking love you dude
Eddie Woo's math skills are so great that he woos many
11:59 how is (k+1)k!=(k+1)! ?
Hi Eddie, nice video! I do believe, in the first proof you need to assume that k is nonnegative, rather than just positive to get it to work. Also to support understanding, I'd suggest putting in equivalence arrows () and implication arrows (=>). This might demystify the thing that "in inequalities you can just chop up and move around", with the emphasis on that in inequalities we only need implication, that clarifies :-) Keep up the good work!
Really grateful.It was really hard topic to understand and thanks to you I'm finally starting to understand it.
Thanks for this high level explanation. Unfortunately, I haven't worked for a strong base in my last-year Discrete course, which I need today in my Data Structure and Algorithm course and in our CS foundation in general. I can make them as my snacks as they're interesting videos. Thank you so much ^-^
how do i know i dont mess up the last inequality with that "substitution"? (k+1)! > 2 × 2^(k+1)
what happends at 12:00 where is the K from the LHS
Who would've thought that speaking good English, being enthusiastic and explaining stuff well would be helpful; couldn't be my professor...
So since 4 is better than 2 you can just say 2? I know you have answered this but I just wanted to ensure I understand ha. Sorry to dwell on it! Thanks again. It would be awesome if you did some recurrence relation vids! Assuming you haven't already! Take care hero
but in Q1 the equal part of the inequality won't work right because yes 6k+3 is necessarily larger than 2k + 3 so if 3^k+1 is larger or equal to 6k+3 then it is larger than 2k+3 but definetely not equal to 2k+3.
Algebraic trickery! Thanks for making a hard subject understandable!
The factorial question actually does work for 4:
RHS=2^4=16
LHS=4!=24
LHS>RHS
I know I'm about 7 years late, but for induction to work, k also has to be able to be equal to zero
So 6k is greater than OR EQUAL TO 2k, which still works, just some of Eddie's working is slightly wrong
I dont know if this is a stupid question, but what if we want to go from 3^(k+1) back to 3^k, we would have to divide by 3 right?
But then the right side becomes 2k + 1,
Which is less than the original, or am I not seeing something?
These aren't too bad. But i feel like recurrences are harder. can you do one where you have to prove a recurrance inequality? like
a_n-1 + a_n-2 >= 2^n (probably false haha, but just an example)
Thanks!!
omg after so watching so many inequality videos and not understanding I finally find yours. Thank you soooo much, you really do know how to explain
So helpful, my maths lecturer at the university of Liverpool is terrible at explaining this you made it so simple, thanks!
+Aidan Sullivan Small world. I'm a fresher there now doing Computer Science and now I'm here trying to learn it, haha.
Watching from 2020. I don't understand anything from my online classes in uni. Now I am here and I understand everything. Thank you!!!
Omg
Thank you
My teacher could never 👏🏻👏🏻👌🏼
And there's me, learning this from my real analysis course b/c I've dealt with so many professors unable to explain this process.
You put this up for free viewing! Amazing. Thank you. I will tell people in my class about this channel.
I wish my professor would lecture like this guy instead of just reading the slides.
Excuse me, I dnt understand i the first example how we got (2k+3) on the LHS. Would you please explain!
We get that because he is trying to prove for k+1, instead of just k. Then 2k+1 becomes 2(k+1)+1. Distributing 2(k+1), you get 2k+2. So once you distribute you have 2k+2+1 which is 2k+3.
OMG thank you so much! Your teaching is so clear and simple to understand :DD
i was wondering if you could do videos on discrete mathematics modules.. such as Relations and Functions.
This explanation is so much better than the one my teacher gave me!
Thank you very much for uploading this :)
why cant my university have you as a lecturer.. i would actually enjoy the course in that case..sigh
Thank you. But K+1 shouldn't be that it's multiplied by at least 6 why did u say 5?
Any 2020 year 12's here after the extension 2 text did a shit job at explaining it
Thanks so much for a great explanation! Finally understand this!
Sir, in the last example, how did the k disappear from the left hand side?
If you mean on the right side of the board then:
In example:
k! = 1*2*3*...*k
Therefore:
(k+1)*k!=1*2*3*...*k*(k+1)=(k+1)!
Thanks
omg he is so amazing this guy is the best teacher i have ever had in my entire life omggg
shit i subbed on the right side, and ended with 3+6k >=2k+3 for the 1st one
You sir deserve more views. We have our Math finals about Pre-Calculus topics and my friend suggested that I go here. I'm not disappointed. Thanks for making induction easier and upload more :D
Very helpfull, Thanks a lot Eddie !
im sorry but at 4:59 isnt it 3^k+1≥1+2k+1-> 3^k+1≥2k+2
In the second example, n!>2^n actually n can be equal to 4 here, not just greater than.
As 4!=24 which is bigger than 2⁴=16, isn't it?
HUGE THANKS!!!!!!!!!!!!!! :))))))) finally got it.
Some teachers are a living proof that no concept is hard
sik tutorial m8
Fizz :D
Thank you so much for this video! Especially the problem with factorials contained in inequalities- I was so confused on how to solve them until I found this video- Keep doing what you're doing please!
Do you have any videos on any of these Counting Principles, double counting, subsets and permutations, partitions, generating functions, derangements and principles of inclusion & exclusion?
I love the energy he puts in the presentation
But this inequality also works for 4
hey eddie :) its hajra from pakistan. Student of 11th grade.... wanna ask a thing... why did you take 0 for n in first example? do we take natural numbers for the value of n?
***** kia?
mishal from pakistan nice to see here you can also take one because we just have to test is that work for n number or not
Dude this is so good. THANK YOU
it don't work for negative number
very helpful, thank you.
damnnnnn.....my uni's discrete teacher is nothing compared to you....can't teach well those tough stuffs...you saved me
why is it 2k tho? where did the 2 come from?
12:28
I don't understand why multiplying my something bigger than 2 (in this case k+1 which is at minimum 5 allows us to substitute)
Let's say
10>2(3) is true
but
10>5(3) is not
So I don't understand your logic. Any help appreciated
***** Hi Eddie, this is a great example of a method for finding these proofs. The part that I get hung up in is with the algebraic manipulation of inequalities. Would you know of any good online resources that provides an extensive review of those principles?
This helped me so much thank you!! From Ireland 🇨🇮
These explanations are excellent. The ones from my textbook are weak compared to the ones used by the professor in this video. Thanks a lot for uploading this video; it was very useful.
7:30 There is a mistake, *k* should range on non-negative integers {0,1,2,...} - it has to include the base case *k* =0. This also means that 6k > 2k is wrong (try to plug in zero), it shuld be 6k >= 2k.
But if it's given k is a positive integer, it's obvious we start with 1. So turns out 6×1>2×1. And then it goes on for 2, 3, etc.
what happend to k! on assuption step 2
Eddie Woo @13:12 how does (k+1)*2^k become 2*2^k... basically how does (k+1) become 2?
Since k > 4 at the basis step, I understand the inequality stands if you decide k+1 is equal to 2... but do you decide this solely based on your assumption (goal)... because for the induction steps, if we were to compute them, then k+1 would start at 6... anyway I think I just answered myself with : "the inequality stands if you decide k+1 is equal to 2" and "you choose =2 because that's where you want to be heading for your proof".
I really appreciate this video! I am a cs student and was having some difficulties. These really cleared it up! If I may ask though... On the one with the factorial, the reason you can exchange the (k+1) with 2 is because you know that k+1 is > 4 so it must be > 2? So at this point you can make that substitution?
Yeah, I don't like that argument. We know k+1 has to be at least 5, but then he weakened it by replacing it with 2.
how does the question say 3^n >= 1+2n and then he switches up and tries to prove that 3^k+1 >= 2k+3. Like where does the +3 come from lol.
when you prove n=k+1 the 1+2k becomes 1+2(k+1) which = 1+2k + 2 = 3+2k
Thank you Mr. Woo, I can now finish the rest of my homework. The inequalities always confused me because I forget you can substitute k+1 with is lowest possible value.
You are really good in your art of teaching. I enjoy your videos from the time I found them. I would like to find out if you uploaded anything about a topic called Extremwertaufgaben in german. I do not know what that is in english. I think I would understand if I learnt few tricks from you.
***** I looked up the link but the work looks so different from my work. For example,a cuboid with a quadratic area is made out out of a 36cm wire.Calculate the dimensions,in which the volume of the cuboid will be at its maximum. Sorry,I sort of translated it from german .Kinda hard.. Thank you..
This has been the most straightforward explanation! Thank you Mr. Woo, you da real mvp
best fucking teacher ever !!!!
That's what he does different his quality where the last point in the prove where everyother video fails to explain that bruh "This is a bit hard to get around ,Yeah this is not a big deal for me but I know it is for you it was for me when I started out ,I FEEL YOU " just that explanation makes the change ( Those little but really valuable tiny details)
@11:54 seconds how did you assume that the LHS was set ? please you may clearify that for future student. Thank you in advance.
In the first example, the induction step needs to work for k = 0 too (not just k > 0) if you anchor at 0. Fortunately it does, though 6·k > 2·k weakens to 6·k ≥ 2·k.
3^(k+1) GE 6k+3 GE 2k+3. Statement RTP follows by PMI Q.e.d
Hey Eddie
I was wondering if you could take your time and help me out with this question
we have to prove that 2^K+1 > 1 + (K+1)*2^k
Thank you so much, I had so much trouble with that one step (going to show that if 6k+3 is less then 3^k, so is 2k+3.)
Hey man. Check this out. Prove the following by the PMI: (1+x)n ≥ 1 + nx + [n(n-1)/2]x², x ≥ 0.
Following your video description wouldn't another way to think why inequality is more flexible than equation simply because it is easier to assume things are unequal than not equal and that equality formally requires two implications to prove (if you think of it as a biconditional) and that inequality only requires one direction. Food for thought and double checking for myself. Great Video. Cheers!
The first example also comes directly from Bernoulli's Inequality. But thank you Mr. Woo for teaching these examples! 😄
THANK YOU EDDIE CLUTCH!
Hi, how would you prove (x+y)^n >x^n +y^n if n is a positive integer?
I'm really glad that you showed how to reason the RHS on the factorial example at the end, but for me that seemed harder. If we multiply both sides by 2, then it satisfies our RHS of 2^(k+1) and gives us 2k on the LHS. Certainly, it follows that (2k)! > (k+1)! since 2k > k+1. That's how my brain worked, but thanks for showing the other way!
what the student is asking at 13:59 ?
Thanks! My Math-Specialist teacher apparently decided we didn't need to know this, despite it being in our course outline...
I can't thank you enough! Induction is still the devil to me, but you've helped me grasp the concept of inequalities so much! our teacher just gave us the chapter to do without explaining anything beyond just simple induction THANK YOU.
why do u work with Z+ n not with N ?
@12:38
You say "If you are multiplying by k+1 then you are multiplying by AT LEAST 5.
However,
k > 4 which means k itself is at least 5.
Therefore:
k + 1 is AT LEAST 6.
Is this correct?
Although I do realize that it doesn't change the point that k+1 > 2 lol. Still works. But still haha.
What you say is true, but it is also irrelevant. This is because k+1 only has to be bigger than 2.
Thank you. It is relevant to my understanding. Making sure I didn't miss something. True none the less.
k must be greater than 4 ,meaning k can be 5 orgreater
The Math God blesses us ! o/
This video is the coolest. I do not know how to give 10 thumbs up, otherwise, I would do it.
morning sir... it is so good to follow your videos but i have a quick question , why do we have to add 1 after n? looking forward for your reply
you are not only a lifesaver but the best maths teacher alive. i am just wandering, which school do you teach at?