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).
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.
@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.
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.
@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"
