The well ordering principle is that every nonempty set of nonnegative integers has a least element.
For a set of nonnegative rationals, shouldn't the least element be zero, or whatever number is closest to zero?
For a set of negative integers, shouldn't the least element be the negative number with the largest magnitude? Why can't the well ordering principle apply to these sets?
Suppose P(n) satisfies conditions (1) and (2) of mathematical induction. Let S be the set of all integers greater than or equal to x for which P(n) is false. Suppose that S is nonempty. Then S has the least element, say, x and so P(x) is false. But by math induction, P(x) is actually true. So S contains all integers greater/equal to x for which P(n) is true.
Does this argument make sense? Thanks.
Are we assuming well ordering principle in this proof of uniqueness of prime factorization?
Suppose prime factorization is not unique. Then among all the integers that can be factored nonuniquely we let s be the least integer such that s = p1p2...pm = q1q2...qm? Is well ordering principle implicitly assumed here?
Question is found here: http://imgur.com/JYtnu5H
If we proved part (a) in this question, is there any work left to do in part (b)? In part (a) we can prove the four squares adjacent to the central square must also contain n, which is essentially saying each square will contain the same number in an infinite grid right?
This is a reading from MIT open courseware. Just in the beginning of section 2.1, the writer states: "the fraction m0/n0 cannot be written in lowest terms, this means that m0 and n0 must have a common prime factor, p>1. But (m0/p)/(n0/p) = m0/n0".
If you want the context, please click on the link as it is written better there than I could put here, but the question is just about the part I copied. I'm confused because my understanding is that if a fraction cannot be written in lowst terms than the numerator and denominator don't have any common factors. The author says they must have.
He then says that any way of expressing the left hand fraction in lowest terms would also work for the right hand fraction, and therefore the lefthand fraction also cannot be written in lowest terms. He had prevously assumed that m0 was the smallest number in the set, but now he states that m0/p will be smaller than m0, which contradicts his previous assumption. I think I can follow this part.
But then he states that since m0 was not the smallest term, than his assumption that the set was nonempy is wrong. This part is also confusing to me. Couldn't it simply mean that his assumption of m0 being the smallest number was wrong and that the set is still nonempty but now has another term smaller than m0?
I'm sorry if I couldn't put my questions right. I guess I really am confused. Feel free to ask me for clarifications and I will do my best.
So number theory thus far is kicking my ass and I just cannot understand or formulate a proper arguement. My question is,
Let Q_>0 = {x∈Q: c>0} be the set of positive rational numbers. Show that Q_>0 does not satisfy the well-ordering principle: i.e., that it is not the case that every nonempty subset of Q>0 has a least element.
My thoughts: Doesn't every set have a empty set contained in it? Wouldn't this show that it is not part of the well-ordering principle then or is there more to it then this? I'm at a loss!
I'm using the Well-Ordering Principle to prove that the following is true for all real values r != 1:
1 + r + r^2 + r^3 + ... + r^n = (1 - r^(n+1)) / (1 - r)
One of the steps we learned for this kind of proof is to assume that the theorem is false and say there is a set of non-negative counterexamples, C.
I understand why the counterexamples have to be non-negative (to use Well-Ordering and find a contradiction about the smallest element in the set), but isn't this overlooking the fact that the theorem could be false and the only counterexamples just happen to be negative numbers?
I realize this is easy using the PMI and that it is the equivalent to WOP, but i need to prove using WOP.
Just to make you guys understand my position, I think pre-ordering is really fucking stupid:
- there's no need to reserve a copy because they are limitless, there's no risk of them running out;
- it's stupid risking buying a game that we don't know how the actual gameplay is, without reviews from our trusted reviewers. The only gameplay we have access are gameplays made by the developer/publisher itself, which is conditioned to look better than it actually is (this also happened to Witcher 3 at one point);
- it's incredibly dangerous following the hype of any game, Cyberpunk 2077 included. People can expect more than the game will be able to deliver, and if we don't put our expectations in check, we will be disappointed.
It's fine if you guys disagree with any or all points I made, I just wanted to make you understand where I am coming from.
So why am I doing something really fucking stupid by my own admission? I want to give CD projekt more relevance. I want to send a message that I want a full premium game for 60 euros. I want to support a studio that actually gives a damn about their work, that doesn't use stupid "gamefied" microtransactions (aka loot boxes).
It would be really easy for them to go the greed way and make a ton of money with loot boxes, all the while hiding behind the excuse that it's too expensive to make games and they need to make money somehow. They give a shit.
And that's why I'm pre-ordering their next game as soon it's available on GoG.
Well.ca has TONS of stuff for our kiddos and at least we are supporting a Canadian company rather than Amazon or Walmart. I know there’s other Canadian companies as well, but Well.ca has a huge selection of things like bottles and diapers, and often they have sales or promo codes. Free shipping after $35.
Hey math folks,
I'm having trouble doing the end of this problem. Here is the problem.
For A: Tsub1 = 1, Tsub2 = 3, and Tsub3 = 5. I believe those are correct.
For B: Tsubn = Tsub(n-1) + 2Tsub(n-2). I believe those are correct as well.
For C:
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Assume nonempty set C that contradicts that the equation is true.
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Assume it has a smallest value m. I made m =4 because I can prove through plugging in that 1, 2 and 3 work.
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This is where I get stuck. In my head I already proved it works because (m-1) is 3 and 3 works. But I'm pretty certain thats wrong.
Anybody have any tips on how to thing about this? Any help is appreciated!
If you want to see my economic principles in a 30 minute animated video, see "How the Economic Machine Works" and if you want to see my Life and Work Principles in 30 Minutes in the same format see 'Principles for Success". And if you want to know "How and Why Capitalism Needs to be Reformed" read my thinking here. Btw, I love ocean exploration which I support through OceanX.
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Proof: https://i.redd.it/fr5k7o1q6pw21.png
Had a great conversation on my AMA today! Thanks for the great questions: https://twitter.com/RayDalio/status/1125886922298204160
I'm investigating the ordering principles and I'm having trouble understanding this one. I have read various sources of information including ching's book, pdf's, websites,etc. I understand that datum is a way of linking/do a set of things, but I still have a few doubts about it:
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1.- ¿With which elements are you supossed to make the datum? ¿With elements of the buildings, like, archs, gable roofs, etc?¿Can it be things like texture or color? ¿Can it be the location of the buildings, maybe if they are located in something like a grid? ¿can it be elements like the roadways or sidewalks of a place (Things that are not buildings, but elements around them)?
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2.-¿ How is it different from rhythm or repetition?
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3.- ¿Is it necessary that the linked elements are similar or can they be different and be linked by placing a perimeter?
