Vice
Proof and Proving Lyrics

We have lyrics for 'Vice' by these artists:

AGE-OF-EP 始まりの合図、聞こえているだろう? 賽は投げられた TVに作られた理想主義者なんて怯えているだけさ生まれた理由(わ…
Anonymuz You didn′t think you could forget so easily, did you? I…
B-xentric I got a uh We all got vices... Jag, Tres Reyes That. Boy!…
Blessd & Pirlo Par de coins, saldo con los tigre' La ambición por entrar…
Blunted Vato Dame dos papeles para yo quemar Tengo un par de…
Brothel You know I'll find you dead or alive But I'll be…
C. Vogt & Patrick Jeremic Izzy Questioning my conscious while I'm gripping on this off…
Chevy Woods Tgod Miami Vice Uh Top off speeding, they say I'm the bad gu…
Chevy Woods Feat. Juicy J & Wiz Khalifa TGOD miami Vice, uh Top off speeding, they say I'm the…

chew-z Vice city Mice a many Where honeys rob for pennies Flashing…
Dina Nan, nan, nan Nan, nan, nan Je sais qu'on a tous les…
Emm Oh I hungered you When I hungered truth Then I prayed to…
EX-ANS Got no time to hold a hostage Even less time to…
F.O.V (Purple Drink Intoxicating music) (Purple smoke and purple s…
Fair to Midland Mountains of molehills, A grapevine in my ear. Spots on the …
G-PAL Vs. Anna Maria X Yeah Shit don't change I'm still in my head I'm still in…
Grandmaster & Melle Mel Vice You have the right to remain silent Anything you say ca…
Grandmaster Flash Vice (4X) You have the right to remain silent Anything y…
Grandmaster Flash & Melle Mel Vice You have the right to remain silent Anything you say ca…
Grandmaster Flash & The Sugarhill Gang Vice (4X) You have the right to remain silent Anything y…
Grandmaster Melle Mel Vice You have the right to remain silent Anything you say ca…
Grandmaster Melle Mel & The Furious Five Vice (4X) You have the right to remain silent Anything y…
Grandmaster Melle-Mel Vice You have the right to remain silent Anything you say ca…
HaVaL Från dem fattiga trakterna Bland dem kaxiga blattarna Häromk…
HDX Sous les lunettes je suis pété Oh je me rappelle ce…
Karma Wears White Ties Roll the dice, let′s see how far you'll push my…
Law & Order: UK I don't know how to deal so I lie And I…
Like Can’t you see the fury In our burning eyes Soldiers report f…
Lincoln 6 Echo This love we share could kill us both Under the skin…
Lisbon First of all, I hit the wall Addicted to addiction Thinking …
M. Flanigan Boys like us don't last that long We don't turn into…
Maria Ahlgren Steady as a needle dropping on a vinyl Neon singer with…
Mega-Alpha I just drove home in an uninspected trash can Listening to…
Melle Mel [conversation of Busy Bee arguing with a cop about a…
Melle Mel & the Furious 5 Vice You have the right to remain silent Anything you say ca…
Melle Mel & the Furious Five Vice (4X) You have the right to remain silent Anything y…
Miranda Lambert Steady as a needle dropping on a vinyl Neon singer with…
Omar Badillo How manny times I saw you Walking out my life? Im…
P.A. Presents 2 Years since my Grandpa Sunny met his sunset I was…
POP ETC You say I'm on your side I'm trying to help…
R.L. King I ain’t wasting no more time on that lame shit Baby…
Razorlight Wild is the wind that strips away our sins Yours is…
RKCB Hourglass shaped, something like a silhouette Wonders towar…
ROUAGE 褪めた瞳の子供と幸せそうな僕ナイフさえもないの? 約束の終わりに禁じられた覚醒? 瞳を閉じれば戻る? ナイフさえも…
Sampling-Sound.com Can’t you see the fury In our burning eyes Soldiers report f…
Shwan If I was your favorite vice You'd bring me home To make…
Swift Guad - Ouais Narvalo, c'est quoi le jeu là? - Ça s'appelle…
The Painted Roses I got a weak man's disease I got a shakin' in…
The Sugarhill Gang - Grandmaster & Melle Mel Vice You have the right to remain silent Anything you say ca…
The Sugarhill Gang vs. Grandmaster Flash Vice (4X) You have the right to remain silent Anything y…
Thomas and the Work-Men Made it out Into the night We're finally breaking free Gotta…
Thythy Daddy says stop wasting time With these men Heard too many c…
Tino Szn (Tino Szn) Huh Snap on this bitch off the dome, off the…
Tobal No.1 I'm okay talkin' to you but ain't no reminiscing I'm okay…
TRXD Nice Vice, nice Vice, nice Vice, nice TRXD Good lookin′, yo…
Tyrant The fire never dying out... Is the waymark to my destination…
Vice & Valor Apple Bottom jeans bend it over Do the Vice Rock ,…
Wiz Khalifa TGOD miami Vice, uh Top off speeding, they say I'm the…
Zlam Dunk Everything that I see So finely made These city lights Won't…

We have lyrics for these tracks by Proof and Proving:

A Welcome First Verse [B.G.] Is you bout what I brang nigga? Biggety-…
A.N.D.rew Text away excuse. Today I bet I'll find the courage to…
For Austria Dearest friend: I'm going mad looking back at what fun I…
Lobotomy With these words that I just can't get and a look…
No It's Just Different Tick marks etched in a wall, in a closet, in a…
Parental Advisory I feel it important to state that I support the…
The Good News This is a coming of age - a time, period, place…

Where in the Simile Am I? A cogent simile: an architect's building - a source of pride…

The lyrics are frequently found in the comments by searching or by filtering for lyric videos

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01Dear Mother0:29

02Where in the Simile Am I?2:19

03Parental Advisory2:55

PT1M51S

12While Away the Mice Will Play1:36

13No, It's Just Different2:24

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Most interesting comments from YouTube:

Mikael Malmberg

5y↑1

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?

ryan alving

5y↑1

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.

All comments from YouTube:

Ashton Scalise

7y↑720

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

Lilia

3y↑1

@Mesko This isn't someone else solving the problems tho, this is somebody giving you the tools to do that yourself. Understanding how proofs work is important, and you'll spend enough time struggling with actual proofs anyways.
Just looking at the answers isn't gonna help you learn, but having someone explain how to find the answers (or looking at some of them, then trying to apply them to other problems) is.

Ken Katatumba

I had similar intuitions until I read your comment and it reminds me of the same doubts that led me to coming up with the keywords to find this video. I thank the content creator and yourself for helping to visualise some of my doubts 😃😃

Loofus

@Sterling Archer well technically you have to pay for wifi

Gaurav Corner

Nad

Yot Thi

@David mèo

3 More Replies...

Tanishq Sharma

2y↑86

2:37 Direct Proof
5:45 Proof by Contradiction
10:30 Proof by Induction
17:30 Proof by Contrapositive

Danny Tran

2y↑1

Great video explaining the proofs, horrible choice of example as it doesn’t cover anything remotely important.

Adam Ahmed

6y↑120

Man, this is my first and will probably be the only YouTube 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 John

@patrickJMT peep

10:!8 PM
10/20/2022

DSPSET3LAPTOP

patrickJMT

6y↑18

happy to help my fellow peeps out there

Ibrahim Alotaibi

5y↑15

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

PJG Community

5y↑114

Could you also count “proof by exhaustion” as a proof?

Devd_rx

@Achyuth Thouta i consider it the least elegant tho, kind of feels like using cheat code to prove something, but hey, since fundamental algebra is built from induction, we can't deny its worth

Achyuth Thouta

3y↑7

@PJG Community Proof by induction is more elegant and the most reliable proof method I've seen

Achyuth Thouta

3y↑3

@TheSeafaringSaxophone I prefer proof by induction

PJG Community

3y↑3

@Wachowski yeah I don’t remember why I asked this question. Weirdly I do remember asking this question and watching this video

Wachowski

3y↑5

you obviously know the answer if you're asking that question

1 More Replies...

Tursko

6y↑32

I am so surprised you dont have more views. Discrete has been killing me and this really connects some of the dots. Thank you.

Michael Lopez

6y↑128

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.

Chikondi Banda

I don't understand why they don't do that

zadmin

@Tayek Miah mans thrives on negativity

OneDerscore One

3y↑2

@Tayek Miah you used the same insult 3 times in a row...

Thành Vinh Nguyên Tô

You were supposed to know the general idea

Blake Brown

5y↑10

@Tayek Miah Good lord, projecting much?

5 More Replies...

Emily Carroll

4y↑6

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.

Mom Monyratana

7y↑3

Thank you! I had a test coming up this Friday and I hope you will put out more examples.

Terri Lajoie

7y↑4

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!

CaptainWumbo

5y↑1

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.

Claudiappena

5y↑5

You explain better than my math prof. Thanks for making this video

jjessicaiscool

7y↑5

I literally learned this today. Thank you!!

Marissa Hosein

7y↑5

Omg I really needed this video, perfect timing, this video helped me sooo much, thank you sir 🙏🏾

Angel Starfire

6y↑2

Seriously thank you 😊I never knew such things existed!

Grd Ielts and spoken English Academy

7y↑1

sir your videos are very helpful. so I request you to please upload videos on topology as I am preparing for CSIR mathematics

theCJoe

7y↑7

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.

mohar husayn

6y↑4

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

Terreqrue

Had my first class of aylasis for enginers but i was late,this helped a lot thanks man

Andrew Bornt

5y↑3

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.

Hedaya Sharif

6y↑1

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

avatar098

5y↑8

I'm about to start grad school for Computer Science and have decided to brush up on my fundamentals. This video is super helpful!

patrickJMT

5y↑1

good luck!

1122333444

3y↑6

every time school makes me feeling like dying, I come and watch your videos and my will to live is restored. bless you man

Issac

3y↑1

Hello. So I wonder which are advanced techniques? Could you please share books for basic and advanced techniques? Nice video 👍

Jazz M

7y↑36

Damn you read my mind, I got a discrete math test with this on Friday!!!!

Kadir Hizarci

6y↑6

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

BlueJacket

😂😂

patrickJMT

6y↑3

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

patrickJMT

6y↑8

fame and money, hahahahahahaahhaahahahahahahahahahahahah

tiptoe 14

4y↑2

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?

Najma Nazrin

5y↑1

where can i get more example videos? i really like the way you teach

Moussa Al Hamouti

7y↑6

I love your videos. Great job!!

Ae7 Df2

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

Demonwhisper

4y↑4

9:52 I've seen some very elegant proofs by contradiction though. Such as the proof by contradiction of the fundamental theorem of arithmetic :-)

Dylan Welch

7y↑1

Would you be able to do videos on automata? Specifically drawing state diagrams for languages that only accept some strings of 'a' and 'b'

Mikael Malmberg

5y↑1

mrmisspretty

4y↑2

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?

ryan alving

5y↑1

Troy

6y↑1

wow man, you ROCK! what an awesome teacher you are

Aefax

7y↑3

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.

Zay Lo

4y↑1

Can we write exception in proof? If we can't, how do we explain in another way?

Fairly Present

3y↑1

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 ???

Raivat Shah

6y↑2

Amazing video! Thanks so much :)

Robin Francis

5y↑1

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.

Ibtissame Ahannach

6y↑2

You are a GOOD teacher !

jungkooki_bunny

7y↑1

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

Guichuan Tang

5y↑1

Right to the point. Excellent!!

DrSatan

7y↑16

Just in time for linear algebra proofs!

SuperShana97

4y↑3

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 :).

SuperShana97

@Loofus Well, it depends on the countries and levels but the course I took is college level.

Loofus

8th grade?

Dill Pickle

6y↑2

I can not thank you enough buddy for your help.Thanks for your service!

patrickJMT

6y↑1

you are welcome, dill pickle

Tommy Chao

7y↑5

Oh man I had three people come up to me for help with proofs, glad this got uploaded!

Noah Hysi

I love the methods of proof.

NukesONLY

7y↑1

Awesome videos! Wish I found you earlier.

StaticBlaster

4y↑1

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).

Trendy Trenessh

5y↑1

Really good video thanks a lot :)

weirdboo

6y↑121

Who else got lucky taking Discrete this semester? haha

TheBlueBeyond

I prefer it then Calculus

Lucas Crown

You saved me throughout discrete math, so grateful. Keep it up

Isheanesu Vunganai

😥

Ryan Bandler

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

Andrea Torres

5y↑4

mee, how did you do in your class?? This class is really intimidating compared to calculus 2

1 More Replies...

Annie Neema K.M

7y↑1

Sir can u plz put a video on teaching how to find the rank og a matrix

Sai sameer

4y↑1

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

Jocelyn Flores

5y↑1

you saved my life!!! thank you sir <3

JUDICIAL REVIEW FOR JULIAN ASSANGE

7y↑4

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.

Danny Israel

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?

Ramone

Hythloday71 ur not proving anything, you solving for x when y = 0

Brazzcode**

4y↑1

Watching this in 2020, man you are great thanks very much you helped me pass my test.

ARYAN THE GREAT

@patrickJMT need more for contrapositive

patrickJMT

4y↑2

Glad I could help!

Owen Gunning

Same (watching in 2020) :)

Maryam Noor

7y↑1

Plz add a lecture on rectangular corrdinates 3 d sketching

Nana Kwame

Please can you solve more examples for all the proofs you mentioned. I still don't understand

Freddy Palacios

patrick got me thru high school calc, hope youre doing great man!

ARYAN THE GREAT

2y↑1

@patrickJMT proof by induction is used fro natural nos. Right?

patrickJMT

4y↑2

i'm doing pretty well, thanks :)

FRANXIS

4y↑3

I always felt some kind of excitement coming from someone who was about to proof something. Prove me wrong

Adeeb

Assume that a person is excited.
But suddenly the ink of his/her pen dies.
Therefore, the person can't be exited.
It contradicts.......

Rose B

without looking at other proof methods how would i know that 2k+1 is another way to represent and odd number

ムネタ

2y↑1

had to look it up in english since spanish wasn't cutting it, thank you so much

The Last Tunebender

4y↑2

Thank You So Much! Was totally lost before I saw this video!

Loofus

Nice name

Genius Turner

4y↑1

Great job!

joketsu100

7y↑2

This was a good idea. Thanks

Boodie Noonga

I wish my professor teach like you

Fufu Chan

5y↑1

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

Eric S.

Great video thanks!

Brianna Henry

7y↑2

Thank you !!!

How Convert

5y↑1

That is basic and useful

kalef1234

6y↑3

that last one was a bit confusing. How does K+1 being the 'successor of k' imply a and b cannot be consecutive?

CmdrColstel

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.

aeroO bics

5y↑4

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.

Calvin Chung

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?

AintItJay TV

6y↑10

You should put time stamps on your longer videos

Zachary Bean

Don't you hate it when you try to go to bed but end up learning math for 2 hours? I can't believe this just happened to me... I kind of like it.

Fadlalmoula Suliman

5y↑1

very good
thanks so much

Syengo Munyithya

6y↑1

This is really sinking.. I almost floated in any lecture but I now understand.

Bilal Molani

5y↑3

If i am given a question to prove. How can i decide that i have to use one particular proof strategy?

Alexander Townsend

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.

Alexander Townsend

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.

RyleeXIII

4y↑2

What about proof by seduction?

Jared Abraham Cid FDA

where did you get the definitions? Are you supposed to make them or will they be given to you

TechWithTrish

thanks so much. this is very elaborate

Dei Swish

4:50 how is b = 2a? what if a = 80? b =/= 161. can you help me out on tht?

Jeisson Sáchica

7y↑1

Awesome!

Jordan

5y↑1

PatrickJMT the OG youtube tutor!

Logamn1023

4y↑1

Wonderful

Shachar H

5y↑1

The theorem is too simple to show a proof by contradiction, that needed a different example.
Also, in general the proofs you showcased are too similar to emphasize the different thinking involved.

e.g. I'd go about a proof by contapositive this way:
suppose a+b = 2k for integers a,b,k (and withot loss of generality b >a).
then b = 2k - a.
Therefore b-a = 2k -2a = 2(k-a) an even number.
But if a,b are consecutive their difference should be 1, an odd number.

Fabrizzio Orderique

4y↑4

I start discrete math in a week :,) someone please help me lol

Nathalee Zaragoza

you are amazing, love from PH!!!

elliott loverin

5y↑2

what if a = b + 1? the two integers would still be consecutive, no?

iSableye

5y↑1

yes it will and successor of b will be a.

Miguel Hernandez

5y↑1

I have a question abut the contraposition proof... If you negate the antecedent "a and b", would this not result in "a or b"?

ထ Anthony ထ

~(A ∧ B) = ~A ∨ ~B

Soilam Mirza

Sir your video very helpful thank

Leviathan

5y↑2

well, another math class another semester of PatrickJMT, here we go...

patrickJMT

please turn your book to page....

Chuck Zacharry G. Igloria

5y↑2

Can you prove this using a direct proof method: "If mn is even, then m is even or n is even"?

patrickJMT

i'm sure one could prove it without much difficulty

John Adams

7y↑200

Hello Mr. Patrick Sir, how would you like your name spelled on my college degree ?

Lou

Loooool

Kirubel Ayehu

Yo,he is life saver

e p

Wow.👏🏼

Arul Gk

4y↑4

HAHAHAHA

Dennis Bolaños

5y↑6

LMFAO!

Tesfish

5y↑1

great lecture note

John Batchler

5y↑2

The u.s. court system is discrete math

John dM.A02

Ah consecutive numbers, when mathematicians can't simply think how to say "the next number"...

StudyWithJosh

Why say many word when few word do trick?

Brain Gainz

Proof? You want the proof? You can’t handle the proof!

Reck

5 minutes in and it already reminds me of problems I face when programming. Maybe I should take discrete.

JUDICIAL REVIEW FOR JULIAN ASSANGE

Proof by Contradiction ... YOU'RE WRONG ! ... ha ha. Seriously though, great, nobody ever bothers to point out that it only works by appealing to the 'law of non-contradiction'.

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