4 ms·

>the union of two minorities is a minority, and the intersection of two majorities is a majority Is that a typo? Shouldn't it be the other way around?

schoen 9y ago
I think you're right. For example, if you have U={A, B, C, D, and E}, then given minority m₁={A, B} and m₂={B, C}, m₁∩m₂={B} is also a minority. Given majority M₁={A, B, C} and majority M₂={B, C, D}, M₁∪M₂={A, B, C, D} is also a majority. This should be clear in finite sets because |m₁∩m₂|≤|m₁|, while |M₁∪M₂|≥|M₁| (you can only make sets smaller or unchanged by intersecting them with other sets, while you can only make sets larger or unchanged by taking their union with other sets).
- taejo 9y ago
  Note that this is a definition of majority/minority, not a theorem proved about them. It's indeed true that by definition, there are no majorities for finite sets.
  - schoen 9y ago
    I didn't quite understand your reply. Did you mean to say that there are no majorities for infinite sets? Edit: It sounds elsewhere like the whole point of this may be the distinctive, counterintuitive properties of infinite sets (in this case especially under AC), but I'm continuing to be a bit confused by the presentation and context.
    - soVeryTired 9y ago
      The article has taken the word "majority" and redefined it to be something counterintuitive, and only partly related to the standard use of the word majority. A under "normal" usage, the words majority / minority have this property: > The intersection of two minorities is a minority, and the union of two majorities is a majority As the GP points out in his reply to my original comment, you can get this property from >If a vote is held and the Ayes have it, and then any number of voters switch from Nay to Aye, the Ayes still have it Now the author is redefining the terms minority and majority so they have this property: > The union of two minorities is a minority, and the intersection of two majorities is a majority But that's not possible unless you have infinite votes. As GP points out, it's clearest when you take "minority" to mean finite, and "majority" to mean "infinite, with finite opposition".
      - schoen 9y ago
        Thanks!
dogecoinbase 9y ago
Nope -- unfortunately, the voting analogy is a poor way to understand ultrafilters. Essentially, filters (a dual to ideals) are structures that tell you which subsets of a partially ordered set are "big", and ultrafilters are maximal filters. A pathological example that can help illuminate the idea is that of a principal ultrafilter ("on x"), which is all subsets of a partially ordered set containing x as a least element. The elements of the ultrafilter are the majorities, and their complements the minorities -- clearly the union of two sets not containing x will not contain x, and the intersection of two sets containing x will always contain x. Choice gives you the existence of a nonprincipal ultrafilter, that is, a structure on the integers without a specific least element which correctly identifies all "big" subsets of the integers in a way that provides that the intersection of "big" sets is still "big". I'd highly recommend the treatment in Just and Weese if you're interested in learning more.
taejo 9y ago
The other way around is already implied by the previous property. Maybe it helps to consider finite sets. A finite set should clearly a minority, and the union of two finite sets, though bigger, is still finite; similarly, the intersection of two co-finite sets is smaller, but still co-finite. Minority and majority are generalizations of finite and co-finite, respectively. (A set is co-finite if the set of elements in some universe that it doesn't contain is finite).
- soVeryTired 9y ago
  Yep - I see it now, you're right. Thanks for the clarification.

schoen 9y ago

I think you're right. For example, if you have U={A, B, C, D, and E}, then given minority m₁={A, B} and m₂={B, C}, m₁∩m₂={B} is also a minority. Given majority M₁={A, B, C} and majority M₂={B, C, D}, M₁∪M₂={A, B, C, D} is also a majority. This should be clear in finite sets because |m₁∩m₂|≤|m₁|, while |M₁∪M₂|≥|M₁| (you can only make sets smaller or unchanged by intersecting them with other sets, while you can only make sets larger or unchanged by taking their union with other sets).

taejo 9y ago
Note that this is a definition of majority/minority, not a theorem proved about them. It's indeed true that by definition, there are no majorities for finite sets.
- schoen 9y ago
  I didn't quite understand your reply. Did you mean to say that there are no majorities for infinite sets? Edit: It sounds elsewhere like the whole point of this may be the distinctive, counterintuitive properties of infinite sets (in this case especially under AC), but I'm continuing to be a bit confused by the presentation and context.
  - soVeryTired 9y ago
    The article has taken the word "majority" and redefined it to be something counterintuitive, and only partly related to the standard use of the word majority. A under "normal" usage, the words majority / minority have this property: > The intersection of two minorities is a minority, and the union of two majorities is a majority As the GP points out in his reply to my original comment, you can get this property from >If a vote is held and the Ayes have it, and then any number of voters switch from Nay to Aye, the Ayes still have it Now the author is redefining the terms minority and majority so they have this property: > The union of two minorities is a minority, and the intersection of two majorities is a majority But that's not possible unless you have infinite votes. As GP points out, it's clearest when you take "minority" to mean finite, and "majority" to mean "infinite, with finite opposition".
    - schoen 9y ago
      Thanks!

dogecoinbase 9y ago

Nope -- unfortunately, the voting analogy is a poor way to understand ultrafilters. Essentially, filters (a dual to ideals) are structures that tell you which subsets of a partially ordered set are "big", and ultrafilters are maximal filters. A pathological example that can help illuminate the idea is that of a principal ultrafilter ("on x"), which is all subsets of a partially ordered set containing x as a least element. The elements of the ultrafilter are the majorities, and their complements the minorities -- clearly the union of two sets not containing x will not contain x, and the intersection of two sets containing x will always contain x. Choice gives you the existence of a nonprincipal ultrafilter, that is, a structure on the integers without a specific least element which correctly identifies all "big" subsets of the integers in a way that provides that the intersection of "big" sets is still "big". I'd highly recommend the treatment in Just and Weese if you're interested in learning more.

taejo 9y ago

The other way around is already implied by the previous property. Maybe it helps to consider finite sets. A finite set should clearly a minority, and the union of two finite sets, though bigger, is still finite; similarly, the intersection of two co-finite sets is smaller, but still co-finite. Minority and majority are generalizations of finite and co-finite, respectively. (A set is co-finite if the set of elements in some universe that it doesn't contain is finite).

soVeryTired 9y ago
Yep - I see it now, you're right. Thanks for the clarification.