Wikipedia:Reference desk/Archives/Mathematics/2020 June 25

= June 25 =

Nations of Nineteen Eighty-Four
This article has a sentence, which I was going to change before I realised I didn't know the answer...Two of the three states are aligned against the third: Oceania and Eurasia against Eastasia or Eurasia and Eastasia against Oceania. But this isn't strictly true: There are three nations—Oceania, Eurasia and Eastasia—all at war, at some point or another, with each other. Two are always allied against the other, but alliances frequently change; yet even two nations together is insufficiently strong to defeat the third. So, what's the maximum number of possible combinations?At first, I thought nine, but that's surely too few. So the article needs only to have "for example" added to the quoted line above. But I'd be interested to know the possible combinations? —— Serial # 08:31, 25 June 2020 (UTC)
 * You're not missing any. You are asking for the power set of a set of three elements, which has eight in total. However, both the empty set and its complement (the whole set) are out of the question, since that would mean everyone allied with everyone, leaving six. Of the remaining six, each set of two's complement is a set of one and vice versa. So there really are only three possible partitions into two nonempty subsets of the three nations. The only partitioning into three nonempty sets is three singletons of a country each so that's not very interesting either.--Jasper Deng (talk) 08:36, 25 June 2020 (UTC)
 * Ah, right, I think I've got it. So, far from there being a large number of combinations, the only possibilities are:
 * Oceania v. Eurasia + Eastasia;
 * Eurasia v. Oceania + Eastasia;
 * Eastasia v. Eurasia + Oceania.
 * Correct? ——  Serial # 08:54, 25 June 2020 (UTC)
 * Yes. The number of binary partitions of a set of n elements into nonempty subsets is $$2^{n-1} - 1$$, from the formula for the size of a power set (and of course more in general if we allow more than binary partitions).--Jasper Deng (talk) 08:57, 25 June 2020 (UTC)
 * provided that the original set is nonempty; otherwise that formula says there are ${-}\tfrac{1}{2}$ such partitions. --Lambiam 12:27, 25 June 2020 (UTC)
 * Nice one. Thanks very much JD! ——  Serial # 09:41, 25 June 2020 (UTC)
 * Singletons. --CiaPan (talk) 20:40, 28 June 2020 (UTC)
 * I don't think it was ever Oceania on its own against Eurasia and Eastasia combined, was it? My recollection is that Oceania always had one ally and one enemy.  Whichever was the ally had always been the ally, and whichever was the enemy had always been the enemy, even though this answer changed (doublethink).
 * But this was the situation as presented to the Oceanian public, and had no necessary connection with any objective reality. It seems likely that the rulers of all three nations simply wanted this state of affairs to continue, and had no desire to win any of the continuing wars.  In that sense they were all three allied, though they sent their soldiers to die on the battlefield against one another. --Trovatore (talk) 00:12, 30 June 2020 (UTC)
 * As I remember it, there is no contemplation of the strategic importance of the alliances; the only function, as far as I remember, is to highlight the power of the Party to control Truth by controlling the past. Whether any other situation than these two alignments (not only another alignment, but also any of the three states ever not having been at war) has occurred or not cannot be determined, nor in fact if this two-against-one is a feature of the Nineteen Eight-Four world or an accidental configuration invariant during the period experienced and remembered by Winston Smith. --Lambiam 10:57, 30 June 2020 (UTC)
 * Goldstein's book mentions the three superstates propping each other up like ears of corn, so that seems to be some sort of contemplation of the strategic nature. But the book itself may be written by the Party (at least that's what they tell Smith after he's caught).  Basically we have no way of knowing if there even are states of Eurasia and Eastasia.  Presumably soldiers go somewhere and fight, and some of them come back, but it could all be orchestrated. --Trovatore (talk) 17:32, 30 June 2020 (UTC)