DIRECT PROOFS - DISCRETE MATHEMATICS
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- เผยแพร่เมื่อ 21 ธ.ค. 2014
- We introduce proofs by looking at the most basic type of proof, a direct proof.
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![[Discrete Mathematics] Direct Proofs Examples](http://i.ytimg.com/vi/uDJfx4bK3Jc/mqdefault.jpg)
![[Discrete Mathematics] Direct Proofs Examples](/img/tr.png)
Thank you so much for your videos!
The world needs more persons like YOU!
At the end, to complete the proof, you have to factor out 2 from 4kj + 2k + 2j + 1 to match the form of 2z + 1 to prove it is odd.
Tommy Saechao yeah this way makes sm more sense surprised he didn’t write it like that
@@tsgoten you replied after 3 years bet this guy have graduated 😂😂😂😂
@@Omarkareem0 LOL
@@Omarkareem0 You replied after 2 years. Thanks for this comment though OC really did help this make more sense.
@@AaronDGreen I REPLIED AFTER 1 YEAR. BET YOU ARE ABOUT TO GRADUATE
Jesus send me to your page to save my butt in this class! I'm in my last semester of Industrial Engineering and I took this class as an elective with 4 other Engineering courses. I was overwhelmed and I pray for God to help me gain wisdom in that class and out of all the youtube searches, yours was the best one i found!!
thank you so much, having a lay way of explaining things saved me from my midterms
These videos are gonna help me pass discrete math, your video on functions where you explained why x^2 is not onto made me go "OHHHH! I get it now!" after like 3 days scratching my head. Teachers has always said that the square root of n = n, no teacher has ever told me that it it equals +-n which is why it's not onto, since +n =/= -n
Thank you, finally I understand. Your videos helped me a lot...
Just to let U Know.. We Had a Test On the First Chapter...on the Truth Tables..The universe quantification and all that stufff.. I got a 90 on that because of watching your videos. U make it so easy to understand. Thank You. I have a another one on this next Week.
Best of luck!
That's funny. I am watching this video right before an exam too.
Jai Patel what grade/year are you in?
First year University
Jai Patel me too except I go to an early college where I take year 1 university classes
Thank You for this video. This must be really helpful for me in my modules.🤗
I've got an exam of this tomorrow. I'm giving all my trusts to this godlike explanations
You da real MVP. If it weren't for you I wouldn't be passing discrete math
Thank you for this clear explanation!
Thank you Sir Its My Presentation next week :) you made it look so easier
Thank you, you explain everything so clearly
Thank you. It helped for my exam next week.
thankkkk youuuuu x100 you are single handedly saving my grade in this class, much love xoxo
Ur welcome love you too
@@AestheticEngineers kinda sus
@@emilyl3952 lmao
5:52 If the key to dealing with odd or even proofs is the +1 at the end. What about for perfect squares and/or rationality?
2(kj+k+j)+1, we have the form in 2(n)+1, and therefore it is odd.
If a question asks you to prove "if x is an integer, then x^2 + 5x - 1 is odd." Do I take "x is an integer" as part of the statement which I can use as an assumption, or its domain?
If we write proof, if there is an exception for a case like unable to divide, do we include that in the statement?
extremely helpful. thanks!
For the if x,y odd then xy odd, I think one could argue that one step is missing to be a valid proof (I know my prof discrete math prof would have been anal about it by removing a good chunk of the mark there.
for 4kj+2k+2j+1, we to factor out a 2 so we get
2(2kj+k+j)+1
Other than that, nice video. I took discrete math a year ago and I was in need of a refresher.
Could you try to add CC to all your videos? I really helps taking notes while you speak
really helpful vedios. thank you sir
Please do a video doing direct proof using rational and irrational numbers as an example
Hi TrevTutor,
Thanks for sharing your knowledge and teaching us about proofs. I like the way how you explain it. Quick comment: why not using definitions in proofs? In the end you argue using definitions that, for instance a number is odd. Then you need to cast you result, after some algebraic transformation, into 2k+1, with k being some integer. Proofs need to be rigorous and I find it important to point that out from the start. Just my 2 cents.
Kuunjalo ú³g
Thank You!
4kj + 2k + 2j + 1 = 2(2kj + k + j) + 1 which can be written as 2n+1 where n = 2kj + k + j
for the first proof i believe you shouldnt have pulled out the n and that (2n^2 + 2n) is the n you are looking for
This is perfect but u have to mention from any set of numbers u get K, j , n.. for all k, n and j
really cool, thanks man
Thanks for the video it has helped alot👏
Congratulations.
Notification: I would like to inform you that I plan to include this video among the videos reviewed in my article "Analysis of TH-camTM Videos and Video Comments on Mathematical Proof Methods". Kind regards.
I'm here to study but I'm falling in love with these videos hahah. Which really is making life easier
The videos love you too
Would have helped to see the precise form of the proof at the end.
Thank you
it was good thnx :)
Perhaps a stupid question, but for the case of X and Y being odd implies XY being odd, could you say x=2n+1 and y=2n+1, so you get (2n+1)(2n+1)?
That's what I did when I tried to solve it myself, and got 2(4n)+1. I know X and Y could be different numbers, but does that matter here?
If you say x = 2n+1 and y = 2n+1, then you're saying that x and y are the same numbers, so your proof does not consider cases where x and y are different numbers.
Thanks!
thank u 😊
Good day sir! I have a question. Where did u get the 4n²? in ur example number 1
When it’s (2n+1)^2 it’s equal to (2n+1)(2n+1) and to multiple these you have to multiply each term in one parenthesis by each term in the other. So four separate multiplication processes added together: 2n*2n (which is where 4n^2 comes from) + 2n*1 +2n*1 +1*1
when you said definition did you mean hypothesis ?
or do it like
...
x²=(2n+1)²
x²=4n²+4n+1
x²=2(2n²+2n)+1
x²=2m+1, where m = (2n²+2n) in Z
Hence, by definition, x² is odd if x is odd.
I had this exact question on my test!
subject ?
I understand what you're saying but i'm so baffled with the factorisation x2= (2n+1) = 4n2+4n+1 could someone please explain that specific part to me
It's x^2 = (2n+1)^2, (2n+1)^2 can be factorized as a perfect square quadratic which has the formula (a+b)^2 = a^2 + 2ab + b^2. In this case a = 2n and b = 1
(www.wikihow.com/Use-the-Perfect-Square-Identity-As-a-Shortcut-in-Expansion)
if this wasn't a very good explanation ^
watch this video and have the song "Forgotten" by Lorne Balfe (13 hours soundtrack) play in the background xD It's so emotional
Around 7 minutes in the video, is it possible to say that WIthout Loss of Generality, case 2 is invalid since multiplying xy is the same as yx
Yes
TheTrevTutor thanks. Also rewatching this now, if a question says "if x or y is odd" is it foolish to assume inclusive/exclusive
Definition: A collection of definitions used to describe something.
Please number your videos! I'm new to your channel and I can't understand what sequence to watch them in
Yey thanks ♥️
the first proof example seems not completely logical. We may only extract 2 out of the bracket as 2(2n2+2n) + 1, and stated clear that n belongs to real number, meanwhile 2n2+2n has to belong to real number. So that we can say 2n2+2n and n perform the same duty in this situation. Hence, we can rewrite 2(2n2+2n) + 1 as 2a + 1 which a = n. Therefore proved. PS: you cannot prove anything of 2n in this case since there's no definition stated what 2n really is and hows a undefined form to multiply another undefined form to get a form that is defined by the prompt.
Where did you get 4n in the line x^2 = 4n^2+[4n]+1
x^2 = (2n+1)(2n+1) = 4n^2 + 2n + 2n + 1 = 4n^2 + 4n + 1
@@eddyhz2874 Thanks brother I understand it now, have a great week😊👍
What do j and k mean? Why not use x and y there
Trev what is the difference between a corollary and a lemma??
A lemma is like a major step we use to prove a theorem. A corollary is something that follows from the theorem once we've proved it.
Thanks :3
i watched this video 10 minutes before the exam HAHAHAHAH
😂😂😂 same here
the 2n+1 example is actually wrong. It is a odd number, but you factor it wrong. You should factor 2 (n^2 + 2n) +1.
=> 2x+1, where x is an integer, x = n^2 + 2n
2 (n^2 + 2n) = 2n^2 +4n does not equal to the original problem.
@@marionettes7974 No he is right, an odd number is defined by the form 2n+1
so when you factor like he did x^2 = 2(n^2 + 2n) +1, you say let t = n^2 + 2n
so t is an integer because the product and sums of integers are integers
Therefore, x^2 = 2t + 1 by substitution
Which is in the form of an odd number where t is some integer
how did you get 4n?
You get 4n^2 by multiplying 4n * 4n = 4n^2. 4n is then retrieved by adding the combined (2n*1)+(2n*1) = 4n. The +1 at the end is retrieved by 1*1, which ofcourse, = 1. It is essentially retrieved through the distributive rule by squaring binomials, which can be hard to see if you're not familiar with it.
I got confused about it the first time myself but here's how it works. :
(2n + 1)^2 = (2n + 1)(2n + 1) =
(2n^1) * (2n^1) = 4n
hi . can you explian how 4kj+2k+2j+1 become ? thank you
become what?
the answer
Hi, I don't know if you still do videos or not. I found your explanations to be very simple and clear and wondered if you could show me how to do the following proofs:
(1) let lcm(a,b) = l so l=pa and l=qb
Prove that gcd(p,q) = 1
(2) if gcd(a,b) = 1 prove that gcd(a+b, a-b) is either 1 or 2.
(3) Prove that if a | (bc) and gcd(a,b)=1 then a|c
(4) If d=gcd(a,b) and f is any other common divisor of a and b, prove that f | d
Any help with any of them is appreciated. If you don't do this anymore no problem. Thanks for all your videos.
this video actually taught me proofs
For the first example, do I have to prove it the same way you did? I simply initialized the variable x as an odd number (3 in this case) and then proved that x^2 (3^2) was indeed an odd number and it equated as true. Is that okay?
Zepheriah You can’t use specific examples to prove a general statement.
nice
Man ily 👌🏻
n^2>n+1
is there supposed to be sound
danke
I'm confused about the algebra
Kind Sir how are you? Stupid question why did you use 2n + 1
An even number is defined as c = 2n, for any integer n.
An odd number is defined as d = 2n+1, for any integer n.
You can see that if n starts at 0, c produces 0,2,4,6,8,10,12,14...
If n starts at 0, d produces 1,3,5,7,9,11,13,15...
+TheTrevTutor is that a correct proof?
+the_path A correct proof of what? These are definitions of odd and even numbers, so they need no proof.
can someone help me with my assignment?
pprove that if n>0 and 4^n -1 is prime then n is odd use direct and indirect method to show that the proposition is true.
Hi please how is it possible to have x square = (2x+1) square why is not 9 instead of four please am a beginner
In French a hypothesis is something to be proved and once proved it becomes a theory
Another stupid q. please bear w me. cant we use n+1 instead of 2n+1?
+Xavaxify Then it'd odd for some n and even for others. So it doesn't work for all n.
Oh, that's right!! Thank u for taking time to reply! :)
+Xavaxify An even number is that it is divisible by 2. This is only if when divided by 2 it leaves an integer/ whole number. 2/2 is even because it =1. 1, 2, 3, 4. 3/2 is not classed as even because dividing by 2 doesn't give a whole number.
Therefore if any even number is divisible by 2 we can write any even number as 2n. n is any number 1, 2, 3, 4 etc. 2n/2 = n where n is any whole number. Since any whole number can be written as 2n. any odd number can be written as 2n+1 or 2n-1.
He has just used this rule to show prove an odd number squared is an odd number. He set x = 2n+1 which means it's odd. Then he squared both sides. at the end he could have factored to 2(n^2 + n) +1 to make it clearer that it's an odd number. (n^2 + n) = any whole number, we don't know the value but it doesn't matter because 2 times any whole number + 1 will make it odd.
+Levi Titan xava actually even +1 = odd so make it sure n is even we multiply with 2 ok in any case 2n can not be odd 2n+1 will always be odd
Thanks for the clarification! :)
ur confusing me... ;_;
have my kids trev
I'll try my best to carry them.
man 😓
hey, what the fuck?
Dear god. Do you actually enjoy doing this stuff? This just seems so damn tedious that it makes it not fun.
Ok so i think trev tutor is good and all but i dont think hes enthusiastic about math. Hes videos are ok but i dont think he gives u good feeling about the depth of the subject matter. Its just meh. Ud probably forget wat u have just watched after a week or two.