# Lec 4 | MIT 6.042J Mathematics for Computer Science, Fall 2010

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# Lec 4 | MIT 6.042J Mathematics for Computer Science, Fall 2010

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90 thoughts on “Lec 4 | MIT 6.042J Mathematics for Computer Science, Fall 2010”

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Career In Forensic Science

haha, i started to take notes as if i was in class ( i forgot that i didnt need to copy this down) .

This is amazing, it is very clear, i can go back and rewind and check if something wasnt clear. I LOVE THIS !!!!

The only diff b/w this and college is that we don't get a degree on paper.

MIT- ♥♥♥ 😀

Man, Tom Leighton, the other lecturer for this class is so much clearer at explaining; this is unfortunate.

I object to your statement in this matter. However, I do agree MIT videos are of immense help.

The reason why I object to your statement is that I have two issues regarding his lecture. First is his accent. This is not a mockery but his pronunciation of words are misleading. Second, I am oppose to the clearness. I find his demonstration to be an excellent example but when he discuss the theorem it started to become vague.

MIT, you rule. You understand that knowledge should remain free and open. So many universities have their lessons under paywalls or lock and key. Thanks for making the world a better place.

"jerk" 😀

oh gosh, his accent is so subtle!

If you'd sat in on some classes I've been in you'd understand just how dense lecturers' accents can get!

He should have excluded the case m=0 at 0:09:46

I college sucks balls

I could not get the proof by induction at the very beginning.

Generally, the steps to be taken for proof by induction are:

1. Inductive basis

2. Induction Hypothesis

3. Inductive Proof/Conclusion

I got the point that m | 0, m | a, m | b, m | x, m | y, but what about m | x+y , m | x+y -a, m | x+y-b ???? :

@Abhay Pai

Without being formal: if x is a bunch of m's and y is a bunch of m's and a is a bunch of m's, then any linear combination of those 3 is a bunch of m's.

So a bit more formal now: There exist i,j,h so that m.i = x, m.j = y, m.h = b.

Now examine:

x + y – b = m.i + m.j – m.h = (i + j – h).m => m | x + y – b

I have always wondered who created the reference angle and why they created it. I've asked many people about this and no one seems to know.

This professor's lecture is speaking the truth. If you are a little bit lazy at a late night to grasp the complete lecture it doesn't work unless you go back and back twice and thrice. It's asking me for my full consciousness (and attention). What can I do more than that? watching in a desktop where we are facing 12 hours unavailability of electric power all over the day and it's only possible at a late night.

can anyone tell me what the name of movie that the professor show his students is?

I learned a lot; thank you for uploading the video and thank you, professor.

Visit my channel for complete Numerical Methods Tutorials (37 videos).

this accent…

aaah die hard 3 ^_^

我居然看完了。这个就是建模吗？

This is my try for this problem: http://allenlipeng47.com/PersonalPage/index/view/160/nkey

dat windows media player on windows xp…

An amazing lecture! Thanks

Is there an answer sheet for the problem sets? The survival of humanity 25 years into the future may depend on this.

The lecture is disorganized. Like, he jumps from one thing to another, without ever completely explaining what the thing was about, and how does it link to the next one (He never proved the corollary at 35:27). I have to do more into understanding, and putting things in place. But still, once I did get all things right, I think he was pretty great. Thanks MIT 😀

about theorem 2, he only proved that the remainder after executing the algorithm is a linear combination of a and b, but he didnt prove that any value between 0 and b can be reached.

i think he also missed one assumption in theorem 2, gcd(a,b)=1. if we take a=3 b=6, then 0< L=5 < b=6 cannot be reached because five is not divisible by 2.

bad lecture from MIT !

Can someone explain me what was 'm' in the original example (a=3, b=5) and if it's m=1 then why didn't the next example (a=33, b=55) allowed m=1 by this logic?

(1) Fill the 5-gal tank

(2) Transfer 3 gallons from the 5-gal tank into the 3-gal tank

(3) Empty out the 3-gal tank

(4) Transfer the remaining 2 gallons from the 5-gal tank to the 3-gal tank

(5) Fill the 5-gal tank

(6) Use the 5-gal tank to provide the remaining 1 gallon into the 3-gal tank.

Your 5-gal tank now has 4 gallons

Somebody should get this guy a hand mirror… :D.

So when we are doing proofs by induction, in general, are we assuming that the entire proposition, meaning the truth value of the operator, ie, the conditional "=>" or are we assuming that both the antecedent and consequent are true?

Because there is more than one way for a conditional proposition to be true (3 in fact), but it looks like we aren't talking about the propositional logic enough to know for sure which case we are talking about in this class.

So for example, "assume P(n) => m|x & m|y." Okay, are we assuming the truth value of "=>" is true, in which case P(n) can be false and m|x & m|y true, or are we assuming that both P(n) and m|x & m|y are true (which also makes the entire "if then; =>" proposition true as well)? We could also assume both P(n) and m|x & m|y to be false and the entire proposition, "P(n) => m|x & m|y," would be true. I mean, I guess it doesn't really matter either way, but it just seems strange to not specify what exactly is happening.

It also seems strange to just use one example of a case where the proposition might be true to prove the entire thereon. Just because it's true in one situation doesn't ensure it's true in every situation. Shouldn't we be assigning probabilities to the strength, read: likelihood of actually being true in every case, of these inductive proofs?

Number theory has the word average that means sums of values/total # of values

What……….. Number theory is a study of integers

I am having really a hard time understanding what 'm' actually is. Can anyone help on this?

(Celebration) praise MIT professors, authorities and others for excellent public education resources, Open Course Ware.

I'll read MIT course notes this summer to help me understand proofs, strong induction, number theory, mathematic notations, logic, applications of math, and other concepts, such as prepositions.

Thanks people.

I appreciate the open courseware.

Where I'm gonna get these handouts from?

I think the problem can be formulated as drawing a star-shaped graph, with “b" ordered nodes, and vertices being drawn from a starting node by moving by “a” to next node… ”a” and “b" must be relative primes in order for all nodes to have connection… Each graph-drawing move is equivalent to 4 bucket moves… Would that simplify the proof?…

Where can I find 18.781 course videos?

Guys help me out here. This lecture doesn't dazzle me unlike the two previous lectures because:

a) Something wrong with the lecturer

b) Something wrong with the lecture

c) I already know where this is going (groups, rings, fields, cryptography, RSA).

d) The combination of above?

I don't wanna sound like an ass, I just don't understand why does it take me 3 days to get through this lecture?

I'll just read the handouts thank you. The Dutch guy is ok but seriously it's not fair to juxtapose him with Leighton.

After reading the mateials, it's more clear and more organized.

His teaching is too vague and hard to understand !

You have to know the bigger picture , Otherwise, you will be confused.

At the end of the course, surprisingly he use a diagram to prove his last theorem. I can figure out what was going on by thinking, but I can't write it down.

The last proof is unconvincing and irresponsible!

But his accent is pretty gentle. 好评

但无论如何他教的还是很棒，比北大的那个强。

这就是著名的中国的辗转相除法。

This is actually a very clever lecture. I didn't think it was possible to introduce state machines, divisibility, gcd's, and Euclid's Algorithm in one problem which looks so trivial on the surface.

This is amazing!

hes teaching him self not us

I don't get why these people give an 1 hour lecture for something that can be covered within 20 minutes.

Hopefully someone sees this post, I need help understanding. If someone could explain where he got the equation sa + tb I'd appreciate it. Maybe its me. My furthest level of college mathematics is Math1B (integral calculus at Berkeley) is that enough that I should completely understand his mathematics?

Someone have some other material that is more clear that cover this lecture?

I think there is an error in the water gallon prove?

are they assuming there is an other empty( with no measurement ) jug to keep the 1 gallon of water?

what will happen if there are 3 or more jugs??

53:20 thats where i got to know m not alone ,,,,the class is also clueless,,,,,,and even so nobody asked a single question

I could keep track of what he's tryng to say ,,,,,,i lost it after 35 mins

thanx anyways MIT

Please help. How do I prove m is the "greatest" common divisor? At 1:11:30, it seems m can be any "lesser" common divisor in those 3 argument.

This teacher isn't as awesome as the other one. Didn't like this lecture. Took to much time to understand. A lot of things are not fully explained. And the logic of proof is confusing. In the poof of Euclidean algorithm he uses the idea that m divides linear combination of a and b, and then proves that m divides linear combination of a and b with Euclid algorithms. Also statement of invariants feels a bit out of the blue. And jug example feels unconcluded. Should have started the lecture with divisibility if the rest relies so heavily on it. Thanks for uploading the lectures, obviously, but I hope the rest of course is as great as the beginning. Coz I kind of had my hopes up.

For those needing the course materials: https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/readings/

at 34:40 he says that if gcd(a,b) = 1 then a,b are primes

but this actually not right as gcd(9,8) = 1

did i misunderstood his words?

https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/readings/MIT6_042JF10_chap04.pdf this is a great reading if it is difficult to follow these lectures.

great lecture but too lengthy 25 lec like these are just too much Ah

Pulverizer sounds like villain from a super hero movie

Is that invariant in the first proof actually just the predicate, and he accidentally wrote it wrong, or am I an idiot?

I have a question about the last conclusion that the GCD is the smallest linear combination of a and b, how do you know every linear combination is included in the Euclid's calculations?

somewhere I see Dr. strang here but these lectures has some harsh comments which I feel is not true and Thanks for making the class interesting with the example

A little bit difficult lecture with the new professor . How many hours does it take to watch this lecture and fully understand it ?

As you can see, professor van Dijk doesn't exactly have the highest ratings, unlike professor Leighton:

https://www.ratemyprofessors.com/ShowRatings.jsp?tid=1902774

Watch lecture 5 first if you need more motivations to study number theory.

Why dont they rub the board before writing?

While you sit for calculating by using a State Machine. The bomb will go off.

I see a lot of negative comments here! And initially I too supported all the comments, but it took me a while to understand that the problem is not with the lecturer, but its with the view. We do not have a view of all the blackboards at a time, and hence its hard for us to refer back and forth.

And the way out to this is, start taking notes of all the things he writes on the blackboard. It will appear tedious in the beginning and will take a little long to complete this lecture, but I am sure you will come back to thank me later! Worked for me, and it will work for you as well. Have fun 🙂

하… 난 멍청인가 잘 이해가안되네… 한국어버전강의없나 ㅠㅠ

函数 f (x) = x

worst teacher at mit ever

https://www.youtube.com/watch?v=1UdFJX8iFrQ. This has some basic animation and explanation for the jug problem. Do check all the related videos to get some clarity on the entire topic.

Minute 54 didn't understand why prof says t'+u diff 0 so r>b.. and why t'+u is equal to 0

at 8:59 the transcript says "define" but the professor said "divide". I die =))))

damm good lecture…..

The number theory course on brilliant.org is really helpful for hammering down some of these concepts. Also cow.math.temple.edu has number theory practice problems.

The previous professor was better.

why so many lislikes on this video?

i don't think there is any problem with the lecture, but I would say I prefer reading the notes over watching the lecture. Unfortunately.

42:43 but in previous class professor said avoid using a word "obvious".

I feel so sorry for his students, pay a huge amount of money and get like this teacher, he is teaching himself not students, at least I'm watching this for free which is good, luckily there is a textbook, 100 times better than this "lecture". But again MIT OpenCourseWare thank you so much for uploading the course you are the best.

This is ridiculous … your would think that these folks are a little more sophisticated !

I am able to understand everything at 2x speed. I dont know why people are complaining?

Happy New Years!

From where to study state machine?

Algorithm at 42:20

https://www.youtube.com/watch?v=H_2_nqKAZ5w

there's a clear explanation about Euclid's algorithm here for anyone interested

@31:46 Doesn't m=1 divide both 55 and 33 (why did we chose m=11). So we should be able to reach anything divisible by m=1 according to the proof. (I know it actually can not. Please point out anything I missed in the proof and not give a "proof by example")

EDIT:

I think I understand why. The theorem proves that the invariant condition has to be true FOR ALL m, s.t. m|a and m|b. So in the example, we need to check for both m = 1, 11.

And then we can conclude if gcd(a,b) divides it then we can reach the state.