Thank you so much. I like how you layout the structure of the proof before we began and then kept mentioning that structure throughout. That really helped to connect everything together.
It took watching several videos, but finally I understand the significance of the term 'there exist/s' . Until now I've been fumbling around with various proofs, and wondering about the process, and not understanding why, when I thought a proof was complete, it wasn't actually complete because there were still parts of the statement to give proof of; And I couldn't work out why it was important to deal with those seemingly trivial elements. NOW, with your emphasis on the word EXIST at least with I understand 'what' needs to be proven, given that I've now understood the significance of the words 'there exists'... Doh! It seems so silly now, but I think I was overwhelmed in class and not paying enough attention. It just didn't occur to me that you have to give proof of all the things that 'exist'. This has cleared up a lot of confusion (there probably didn't even need to be confusion, but, oh well). Thank you.
a bit late but if it states that "there exists", then we have to prove an existence of an object. If it states "for all", we have to prove that the conjecture/theorem holds true whatever the value of that object is(in its set).
@8:28 when we have shown that abs(a) * abs(q - q') < a. isn't this enough for a contradiction because this is impossible? a * positive integer cannot be less than a. I know this video is old but if anyone could let me know if this is also an appropriate way to do it thanks :)
I'll try to help but I think you get it and you're just making a bigger leap than she does. At this point in the proof, she's trying to show that q and q' must be the same integer. She's established that: abs(a) * abs(q - q') < a. Just to clean it up, she's also established that abs(a) = a because a>0 (that was given to us and the definition of absolute value makes this true). So, she's established that: a * abs(q - q') < a. But, the abs(q-q') must be non-negative by definition of absolute value. So, the only way for 'a' (which is greater than 0) times a non-negative integer to be less than 'a' itself is if the non-negative integer is 0. If (q-q') = 0, then a * 0 = 0. This result (0), is less than 'a' because, again, a>0 per the theorem. Again, as the proof shows, this is the ONLY way to get this result. This means that q and q' are the same integer. She shows this using algebra to divide by 'a' and then notes that this shows that abs(q-q') must be less than 1. She then uses the definition of absolute value to also say that abs(q-q') >= 0. The only integer that is less than 1 and greater than or equal to 0 is 0. So, abs(q-q')=0. This means q=q'. And, since the definition of r and r' differ only by q vs. q', this means r=r'. This proves UNIQUENESS. I hope that helps.
@Sophisticated Coherence I wouldn't call it a proper lecture...he just reads something then when he can't explain it he literally goes "I don't know how that happened, but this will be in the test"
Beautiful explanation. But i have a confusion. During the proof of existence, why was it required to prove the existence of two elements? Because to apply the well ordering principle it is enough for the set to be non-empty. So then why prove the existence of two elements? Why not prove the existence of only one element and continue the proof? Correct me if I have mistaken the concept.
Great work! Would you mind kindly telling me what software are you using to create this video of amazing quality? I found it nice to have well-typed math symbols.
proofs feels like magic until someone explains it. thanks for the video!
thank you for uploading theres hardly any videos on these types of subjects. your video is clear and helped out so thank you for uploading!
Thank you so much. I like how you layout the structure of the proof before we began and then kept mentioning that structure throughout. That really helped to connect everything together.
A wonderful--and quite helpful--explanation of this proof. I hope you continue your meaningful contribution to the math community.
I needed this lesson, it most certainly helped. Thank you for explaining every little detail.
It took watching several videos, but finally I understand the significance of the term 'there exist/s' . Until now I've been fumbling around with various proofs, and wondering about the process, and not understanding why, when I thought a proof was complete, it wasn't actually complete because there were still parts of the statement to give proof of; And I couldn't work out why it was important to deal with those seemingly trivial elements. NOW, with your emphasis on the word EXIST at least with I understand 'what' needs to be proven, given that I've now understood the significance of the words 'there exists'... Doh! It seems so silly now, but I think I was overwhelmed in class and not paying enough attention. It just didn't occur to me that you have to give proof of all the things that 'exist'. This has cleared up a lot of confusion (there probably didn't even need to be confusion, but, oh well). Thank you.
a bit late but if it states that "there exists", then we have to prove an existence of an object. If it states "for all", we have to prove that the conjecture/theorem holds true whatever the value of that object is(in its set).
@@Karlthegreat100 Thank you; this is a nice, concise summary.
Absolutely brilliant explanation! It was very concise and easy to follow! Thanks a lot!
thank you so much, I really appreciate the way you connected different steps of the proof and also explained why those steps were necessary
Most intuitive explanation I've seen on this.
THANK YOU!
It is the clearest presentation for this proof i have ever found, and your voice is great btw.
Very structured :-) Easy to follow because you clearly motivate the upcomming steps.
This was absolutely incredible. Thank you so, so much.
Very clear.. Awesome explaination..
Thank you thank you thank you!!! This video was so much more helpful than my professor at explaining this proof!
You made it so clear for me, thankyou very much♥
Thankk youuu I was struggling to understand it. Loved ur way of explaining hope u keep doing vids ❤️❤️
That is the best explanation I came across on UA-cam
Thank you! Slightly different than the proof I am studying, but provides great insight into solving this problem.
Thank you so much!This is beautiful, you're the best
Thank you so much, was finally able to get my head round this, very helpful
@8:28 when we have shown that abs(a) * abs(q - q') < a. isn't this enough for a contradiction because this is impossible? a * positive integer cannot be less than a. I know this video is old but if anyone could let me know if this is also an appropriate way to do it thanks :)
I'll try to help but I think you get it and you're just making a bigger leap than she does. At this point in the proof, she's trying to show that q and q' must be the same integer. She's established that: abs(a) * abs(q - q') < a. Just to clean it up, she's also established that abs(a) = a because a>0 (that was given to us and the definition of absolute value makes this true). So, she's established that: a * abs(q - q') < a. But, the abs(q-q') must be non-negative by definition of absolute value. So, the only way for 'a' (which is greater than 0) times a non-negative integer to be less than 'a' itself is if the non-negative integer is 0. If (q-q') = 0, then a * 0 = 0. This result (0), is less than 'a' because, again, a>0 per the theorem. Again, as the proof shows, this is the ONLY way to get this result. This means that q and q' are the same integer. She shows this using algebra to divide by 'a' and then notes that this shows that abs(q-q') must be less than 1. She then uses the definition of absolute value to also say that abs(q-q') >= 0. The only integer that is less than 1 and greater than or equal to 0 is 0. So, abs(q-q')=0. This means q=q'. And, since the definition of r and r' differ only by q vs. q', this means r=r'. This proves UNIQUENESS. I hope that helps.
Can anybody help me to find how the Set S contains only positive integers, so that WOP can be applied!!!
Absolutely brilliant! - liked and subbed.
thank you so much, ma'am.
You're welcome
Thank you so much for uploading this video...this helps me too much...
Thank you very much but could you pls do examples
Thank you so much for this!!!!! The proof in my test book is needlessly complicated with 4 parts instead of yours having only 2.
Thank you for making this proof so clear.
THANK YOU QUEEN!!!!!!!
Excellent video lecture.
Awesome explanation 🔥🔥🔥
beautiful proven, thanks for sharing
Brilliant . Thank you
You explained this way better than my professor
@Sophisticated Coherence I wouldn't call it a proper lecture...he just reads something then when he can't explain it he literally goes "I don't know how that happened, but this will be in the test"
Great proof, great explanation, but shouldn't we also assume b>a? Therefore we can assume that both a, b are positive integers such that b>a?
Thank you madam.... Please post more videos on these topics...... I am highly helped with this... 😆😆😆
so nice
good video, greetings from bogotá colombia
Where comes b-xa from?
Beautiful explanation.
But i have a confusion. During the proof of existence, why was it required to prove the existence of two elements? Because to apply the well ordering principle it is enough for the set to be non-empty. So then why prove the existence of two elements? Why not prove the existence of only one element and continue the proof? Correct me if I have mistaken the concept.
both sets are different.
We aren't proving the existence of two elements. She was showing that regardless of the sign of b, the set is non empty.
thank you mam this proof is very systematic
Very thorough Thank You
in existence part, for b>0 why do we only put x=0? can we put some other value except x=0?
Vedprakash Meena is right also in that part the main goal is to show that the set S is not empty in order to use the Well-Ordering Principle.
Thank you!
Great work! Would you mind kindly telling me what software are you using to create this video of amazing quality? I found it nice to have well-typed math symbols.
Now I'm clear about the uniqueness part
Thanks for the tutorial
Thank you thank you thank youuuu
thank you so much
Cute Clear Crisp
Thank u soooo much ma'am this was the best one
Let's assume that : YOU are the BEST!
It doesn't hfta be proven cuz it's a definition outta me that you're the best🌹
SIMP
Best video.
thank you so muchh
it helped a lot
Superb thanks
Thanks !!
Thank u so much !
the video was great but can u go a bit slower?
Exxxxxxxxxxxcccccccellllllllllllllllentttttttttttttttt.................I really realised after watching this video..........................
SAVAGE
ma'am
his
hiii