Strong Induction // Intro and Full Example

[@danielerhabor3241]

Easily the best explanation on Strong Induction so far. My test is in an hour. Wish me luck.

[@danielerhabor3241]

Assuming the Fibonacci sequence prove F(n) < 2^n for n = 1,2,3... by strong induction. I think I got the base case but I couldn't algebraically manipulate the inductive case to prove it. I failed you sensei.

[@BADR567567]

MY TEST AFTER 1 HOURE TOO, i will not failed you sensei @@danielerhabor3241

[@danielerhabor3241]

@GeishaTheSerpantClan professor looked at everyone's papers and realized no one was making sense so he just said "just use regular induction". So I survived.

[@mayankjangid1543]

Literally, this is the best explanation ever!

[@nafisaumar8559]

This is me rn

[@DanT-iu6oc]

Are math professors incapable of explaining this concept? Are their memories wiped right before being put in front of a class? I'm paying 100k tuition that a 10 minute YT vid just completely beat out.

[@nolanrudolph5463]

I feel this comment down to my fucking bones dude. Amen!!!

[@uri_1199]

yo got robbed bro

[@F7FD]

LITERALLY. at this point i only go to uni for the degree. all of my "knowledge" is from youtube

[@1Hydraulic1]

unfortunately professors in universities are hired on their research capability, not their teaching ability

[@jake_3745]

im always so flashed when i see a number like 100k for a university... america is doing something wrong

[@enzoporte]

10 minutes to explain what an old university proffessor could not in 120 minutes. Thank you very much for making this video. I hope you know how much you have been of help to many people by putting effort into your teaching :)

[@Luis-tf9pb]

It's in fact the best explanation, with no complicated definitions, just straight to what is really important!

[@FenjaminBranklin95]

You are the type to be accepted at the gates of heaven. Thank you so much.

[@combinedmathsbysachithband5680]

i dont know what kind of idiots dislike this beautiful simple explanation.thank you sir

[@connorkelly865]

Probably all the Profs realizing he's doing their job for free lol

[@jelenavaryukhicheva1197]

That was a much clearer explanation than MIT video I watched before on the same topic.

[@novelas3536]

I love this. I needed a more rigorous approach to this topic.

[@malcolmhollett8638]

Not only did he explain one concept professors can't seem to explain, he managed to explain 2!! Mathematical induction and strong induction... How the ....

[@ymingxi]

This is the best explanation that i have ever had !!!

[@sarahwhite4227]

Wow! Honestly, video was so clear and easy to understand! I went through many textbooks and videos and this is definitely the best one! I am definitely subscribing

[@bizarrapmusic]

Jesus man, thank you so much. It wasn't that difficult or different of a normal induction after all. I don't know why they always try to explain things in the most difficult way possible on the books, i mean seriously, it's like they trying to prove that they can use a bunch of elegant words and things to say "look im smart!!". Again thank you!

[@a_ghoul]

wow you are really good at explaining these things! Thanks. So know the way that I understand it is that for some problems you can't just assume to step of the ladder that you are on, but all of the ladder steps below you that you've already climbed up. Very cool.

[@EricTighten]

God, this is way more clear than my professor's lecture! Thanks!

[@DrTrefor]

Glad to hear that!

[@adamgimbel9128]

Best video I have seen so far! very enthusiastic and well spoken. Thank you sir.

[@StradexEngine]

Thank you for this, I was reading a book about discrete maths and I was having a hard time trying to even understand Strong induction!

[@T0ra99]

Zybooks be like

[@johannashaw7807]

Very clear and easy to understand explanation for strong mathematical induction, thank you so much!

[@kingofgods898]

I'm convinced math professors make stuff like this seem harder than it is just so they can feel smart.

[@charlietran5605]

Dr Bazett, thank you so much! Your explanation is so clear and simple.

[@DrTrefor]

Glad it was helpful!

[@TrojenMonkey]

If this is the best explanation there is, the future is bleak. When you brought out 2p+1 +2(2q+1), I was truly lost. I have no idea where you sprung that from. "We can write them in the familiar 2p+1...", what familiar way? By whom?

[@wm1pyro604]

kzbin.info/www/bejne/mp2ulJl9qpKKpqs

[@galencollins7936]

Thanks- this video is helpful. I understand proof by induction, but I am having a tough time understanding strong induction or knowing when to use it.

[@tomashaddad]

that step where you use the IH is fucking beautiful and made everything click, thanks

[@aashsyed1277]

You a STRONG explainer.

[@PLP0321]

I've watched many videos and this seriously is the best explanation. Thanks again!!

[@hannaranah]

Thanks! :D My professor had me very confused about this, you just saved me a lot of frustration.

[@DrTrefor]

Glad I could help!

[@marcschmidtpujol550]

Great! My teachers first example for strong induction was to show that any natural number n can be expressed as n=r*2^s, with r odd and s natural. I don’t understand why he did that… even he said it might be hard to understand. Thank you very much for the vid!!

[@lihongming1]

This helps so much, thank you! Looking forward to seeing more videos about discrete math from you!

[@Kyle-xk2rb]

Thank you so much. My textbook doesn't cover this very well.

[@boxoftruth2903]

man I love you. Clear and concise explanation cleared everything up :)

[@elischrag8436]

8:42 how could 2a sub k possibly be odd?????? It’s literally being multiplied by 2. Someone explain???

[@Tony-yn5rr]

I HATE INDUCTION I HATE IT HATE IT HATE IT

[@delightfuladebiyi9096]

😂😂😂😂😂😂😂😂

[@fuadbaghirli246]

me to bro

[@mostafakamal430]

This make complete sense and I understand the concept but all the problems that I have to do is so hard to solve. It deals with induction on trees, nodes and some other bullshit stuffs!!

[@slazter274]

I really like the explanation, However u've marked, 2ak as odd, which should be wrong? AK is odd, but 2*(k+1) = 2k+2, which is divisible by 2, therefore even. Or mabye you just meant to drag the bar under it to just mark out the ak part. Anyways, so far the best video i've found on explaining the difference between Regular and Strong Induction,Thanks alot!

[@sanjayabudhathoki2742]

9:35 Just a thought, isn't 2(ak) always even because of we multiply any number by an even number, it's always even

[@ramyhuber8392]

Yes I agree, my understanding is that 2 times any integer (even or odd) is always even. I stopped video here to mull this over. Is this a mistake in the video. Odd plus odd is not always odd, odd plus even is.

[@QuixoticMcGee]

@@ramyhuber8392 I think he meant that Ak is odd because he writes it correctly in the next step.

[@amadif1793]

Such a clear and engaging explanation! Thanks!

[@maanasperi9989]

umm at 8:55 wouldn't 2 times anything be even? Or are we just assuming 2 * ak is odd for the sake of the example?

[@fleetinggreen8565]

I think he's pointing out that a_k is odd due to the induction hypothesis, not 2*a_k

[@chantalespinosa5184]

i just asked the same question

[@ruveydayilmaz4894]

9:28 looks like I gotta pee

[@kat1345]

lMaO

[@henri1_96]

:DDD

[@prrmrr7358]

Thank you so much for such a clear explanation!

[@speedysenu4183]

Wow, thank you. Such a clear explanation!

[@nadaabdulla7556]

I’m sick of number theory, actually I didn’t know that math is like this ..

[@riaparikh1329]

such an easy explanation!! thank you

[@chantalespinosa5184]

how is 2ak always odd, when ak itself is odd? uneven * 2, is always even? 8:41

[@anonymousplayer9776]

he meant ak is always odd

[@steven9492]

excellent video my guy

[@md123180]

Khan: nope. Trev: nope. Patrick: nope. About 30 other confusing videos: nope. Trefor: yup. I don't like it, but I get it now.

[@wangyun9360]

Hi Bazett, thank you for making this great video! I have a question regarding the example on 6:58 about the base cases, why do we need a1 and a2 instead of a1 only?

[@ita6444]

Because the sequence is defined by the two previous values so to to find the next one you have to know the two values that came before. So unless you have two basis values you can't go forward

[@pianofreak247]

Thank you for this. Helped a lot

[@RyanTorrecampo]

Great explanation! Thanks for making this video extremely easy to follow.

[@dnavas7719]

At 8:20 why are the 2 lower levels those instead of "a_k-1 + 2a_k-2" ?

[@dnavas7719]

@@DrTrefor ahh makes sense, thanks!

[@arnavgupta9509]

Great explanation!! Thanks

[@RizqChafikElIshaki]

you are the best thank you so much, may allah bless you.❤

[@davidFbeckham]

Really well explained! Thank you

[@dancho8161]

this video helps me sooo much, thank you.

[@turokg1578]

are there any use cases of this over mathematical induction except recursivie definitions?

[@charlesedeki--mathcomputer7198]

Please what is the name of the software you are using for the video, its great way to present lecture, thank you.

[@aaryaanshaik2103]

you suggest that 2ak must be odd by inductive hypothesis, but if ak is odd wouldnt 2ak be even?

[@CreamyFucker]

I think the brackets are meant to only surround the ak and not the 2 coefficient. In that case the 2ak would be even and the a(k-1) would be odd. An odd plus an even is an odd so therefore a(k+1) is odd.

[@shreyakapoor2002]

Thank god u are here

[@justynfogarty6209]

This was such a great lesson, thank you.

[@user-su7zk3ve7z]

thank you so much!

[@capybara-k6g]

8:45 you assumed that a(1),...,a(i) is odd, not a(i),...a(k). So if a(k-1) and a(k-2) are in the interval a(i),...a(k), then the inductive hypothesis doesn't apply??

[@alextitov1416]

Great video!

[@sofieooms8990]

wow thank you, this was really easy to follow

[@김창성-x2y]

Thank you sir! This helped me a lot!

[@gabeb6834]

Big thanks man!

[@tomificationness]

this helped a lot for graph theory!

[@btobin86]

So assuming that ai is odd for 1

[@selimcanpolat8664]

i don't think we can conclude that consecutive terms a_k-1 and a_k are odd just by looking at the inductive step

[@CalleTful]

Good video! Where does this 2p+1 come from?

[@tomashaddad]

definition of an odd number is 2p + 1 for p in Z, likewise even numbers are 2p for p in Z

[@MustafaBerkeGureltol]

A Dr. looks younger than I do lol

[@adamkolany1668]

Well, we used to call that ordinal induction. Can you show me some sources in literature on set theory (Kunnen, Jech, Bar-Hillel, etc ?) where this "strong" also is used?

[@MagnusTheUltramarine]

So K in this case is any random number between (1, minimum number) and the maximum number (for example: 11) So K could be in this case 9, and i is any number between 1 and 9 and P(n) is Ak = Ak-2 + 2*Ak-1 Is this correct?

[@kenneth078]

what is the difference of that between weak mathematical induction?? basically it is the weak induction with an additional “step 2” stated out of nowhere. Am i missing sth or what? I don’t understand.

[@MuffinsAPlenty]

What do you mean when you say that strong induction allows you to prove a more general set of claims than regular induction? The two methods are equivalent, so any claim you can prove with one can be proven with the other. (Though it can be a lot less obvious how to prove something with regular induction than strong induction.)

[@bradwang3648]

Very helpful, thank you!

[@wm1pyro604]

great video thanks - the only thing I don't understand is what the limit on my k should be ("k is greater than or equal to...")

[@ajakororo5366]

8:36 how is 2 times some integer odd?

[@davidfarah]

He meant the integer without the 2

[@williamgriffith2822]

i love you more than everything bro

[@perturbant1917]

thanks

[@al7aroDos]

Do you have something about structural induction?

[@jean-michelcid9594]

thanks. great content

[@angelaituriaga9179]

How do we know how many base cases we should be proving?

[@umadbroyo2388]

Question: how can you assume a_(k-1) is odd when k can be equal to 1, so k-1 can be 0 and our base case doesnt proof for value being 0?

[@KennethKreindler]

i>= 1 no?

[@seanli6339]

can u make one on structural induction with recursively defined sets

[@benphillips93]

I have n = 7a - 3b where n is greater or equal to 1, any ideas?

[@vinayak186f3]

Induction is interesting ...

[@MO-xq2wm]

Why does it seem like all our teachers are just useless??

[@TDefton]

How is 2a_k odd if the definition of an even number is 2n (where n is any integer)???

[@TDefton]

Oh nevermind I just saw your comment. You're saying the product is a_(k-1) + 2a_k is odd. Gotcha

[@RainyDay_xoxo]

If you assume a_i is odd, for i equal or larger than 1 and equal or smaller than a_k, how can you then claim a_k-1 and a_k are odd? You havent assumed anything about k, right? Only about i.

[@matthewlev1342]

I still dont get the difference between this and standard induction?

[@rocketleague2136]

thank you very much

[@Mahi_013]

Thanks!

[@MakingVidsBreh]

What is your conclusion?

[@loganshawkins962]

Could you make another video you lost me at P(1)

[@civenge1802]

Isn't 2Ak even? 2 times an odd number is an even number.

[@DrTrefor]

Sorry, my underline is meant to only be the a_k not the 2a_k

[@civenge1802]

@@DrTrefor Thanks for the clarification.

[@alonsovega7667]

U must also emphasize the fact that k-1 is between your least element of your set and k.

[@amorfati4559]

Thanks man.

[@nouf9183]

I love you sir

[@VoxScripts]

linus tech tips teaching math

[@raylauffer2224]

Thank you.

[@Rita-rs5qi]

why do some people use k+1 while others use k-1?