Cantor Definition and 72 Threads

A cantor or chanter is a person who leads people in singing or sometimes in prayer. In formal Christian worship, a cantor is a person who sings solo verses or passages to which the choir or congregation responds.
In Judaism, a cantor is one who sings and leads people in prayer in a Jewish religious service and may be called hazzan. A cantor in Reform and Conservative Judaism is an ordained clergyperson, similar to that of an ordained rabbi, if the cantor has gone through seminary training or been certified as a "Cantor" from an endorsed seminary.
"Cantor" is used as a translation of equivalent terms in other languages, such as for the leader of singing on a traditional Kerala snake boat, a Chundan Vallam. A similar term is precentor, defined as a leader of the singing of a choir or congregation.
Cantor is a Coptic word (language that very similar to Greek) which if translated to English means teacher.
More specific types of cantor include:

Cantor in Christianity, an ecclesiastical officer leading liturgical music in several branches of the Christian church
Protopsaltis, leader master cantor of the right choir (Orthodox Church)
Lampadarios, leader of the left choir (Orthodox Church)
Domestikos, leader assistant to the Protopsaltis of the right choir and/or to the Lampdarios of the left choir (Orthodox Church)
Precentor
Succentor
Hazzan in Judaism, a singer and/or musician. Orthodox Judaism only allows men to be cantors, while the other branches allow women. Reform Judaism and Orthodox Judaism ordain cantors from seminaries. Ordained cantors serve as clergy in their congregations and perform all ministerial rites as rabbis.
An ordained Muezzin, who calls the Adhan in Islam for prayer, that serves as clergy in their congregations and perform all ministerial rites as imams.

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  1. F

    Can Venn diagrams resemble Cantor sets with infinitely many sets?

    if you go down to the section called "do they always exist?" you'll find a venn diagram for 6 sets: http://www.combinatorics.org/Surveys/ds5/VennWhatEJC.html would a horizontal cross-section resemble a cantor set, if there were infinitely many sets? it looks like it would vaguely resemble...
  2. A

    Cantor diagonalization argument

    sorry for starting yet another one of these threads :p As far as I know, cantor's diagonal argument merely says- if you have a list of n real numbers, then you can always find a real number not belonging to the list. But this just means that you can't set up a 1-1 between the reals, and...
  3. M

    Understanding the Fat Cantor Set & Proving Its Measure

    What is the definition of a fat cantor set? How do I show that the fat cantor set has positive Lebesgue measure and does not contain any interval. I know for the cantor set that at each stage, we remove the middle third of each interval starting with [0,1]. I am wondering if instead for the...
  4. W

    If C is the Cantor set, C+C contains an open set.

    This is a statement my professor made in class some time ago (as a means to show that C contains a Hamel basis) that seemed fairly innocent, but it's bothered me for awhile. I did some searching online, and it seems that C+C=[0,2]. There it was again stated that this is fairly easy to show...
  5. D

    Distance function from x to the Cantor set

    Does the said function satisfy: (1)continuity (2)never constant (3)has uncountably many zeroes 1 and 3 is trivial, but I'm not sure about 2.
  6. Bob3141592

    What are the endpoints of the interior points in the Cantor Set?

    The Cantor Set is making me very confused. I can understand that since only open sets are removed, the Cantor Set if a collection of closed sets. I believe I understand that the Cantor Set has measure zero, and therefore contains only intervals of zero measure. I can see that the endpoints of...
  7. B

    Understanding Cantor Set - What Are The Points Between Endpoints?

    Let C be the Cantor set Let A be the set which is the union of those end points of each interval in each step of the cantor set construction It seems to be true that A is countable and C is uncountable. Moreover, A is a proper subset of C. But I cannot imagin what kind of the points in C - A...
  8. S

    Cantor set defined via sums, whaaaaa?

    Cantor set defined via sums, whaaaaa?!? problem 19 chapter 3 of Rudin. I'm totally lost, I've even done a project on the Cantor set before but I just don't know where to start here. Associate to each sequence a=(p_n) in which p_n is either 0 or 2, the real number x(a) = sum from 1 to...
  9. H

    Density of [0,1]-C on Thick Cantor Set Construction | Sequence {a_n}

    Let C be the thick Cantor set. let {a_n} be a sequence of positive numbers. In the construction of the thick Cantor set, at the n-th stage we remove the middle a_n part of each interval (instead of the middle third as in the ordinary Cantor set). I actually wanted to show that [0,1]-C is...
  10. A

    Was Cantor Actually Wrong?

    was cantor wrong?? (funny) the majority of digg certainly seems to think so: http://digg.com/general_sciences/Strange_but_True_Infinity_Comes_in_Different_Sizes_2 thought you guys might find this entertaining
  11. P

    Is the Cantor diagonal argument conclusive?

    Cantor diagonal argument-? The following eight statements contain the essence of Cantor's argument. 1. A 'real' number is represented by an infinite decimal expansion, an unending sequence of integers to the right of the decimal point. 2. Assume the set of real numbers in the...
  12. M

    Does f(x) have to be in the ternary Cantor set?

    Hey all, I would really like help on this probably simple proof: That the map x |--> f(x) = (x+2)/3 on [0,1] is a contraction, and maps the ternary Cantor set into itself. Also, find it's fixed point. (1) I can easily show the fixed point (where f(x) = x) is 1. (2) I can also...
  13. C

    Zeno's Dichotomy and Cantor Set

    Zeno's Dichotomy paradox divides the distance traveled by any traveled into an infinite geometric progression. ie: 1, 1/2, 1/4,... and so on. The argument is that the traveller must cover these individual distances before he can complete the whole. But since the distances to be traveled are...
  14. B

    Explaining Cantor Set: Countable or Uncountable?

    Can someone explain how the Cantor set can be uncountable but also contain no intervals? I am assuming that as k goes to infinity, we are left with 0 and 1 in the final interation so the set is finite with those elements. The set of natural numbers is countable so I can bijectively map every...
  15. B

    Cantor set and Base 3 expansion

    How do I do a ternary expansion of numbers, and prove that if a number is part of the 2^k iteration of the cantor set if and only if each decimal expansion position is either a two or zero? If you guys can give me a hint, I would love to go from there.
  16. D

    Exploring the Construction of the Cantor Set in Base 3

    How can I write down a closed form expression for the endpoints used in the construction of the Cantor set? i.e., 0, 1, 1/3, 2/3, 1/9, 2/9, 7/9, 8/9, etc.
  17. JasonRox

    Is Cantor's Theorem Valid for Monotone Decreasing Families in T1 Spaces?

    So, here I am again. This is to only have a proof reviewed. Like I said before, I will do this from time to time, so I know I'm staying on track. Theorem - Cantor Theorem Let {C_1, C_2, ...} be a monotone decreasing countable family of non-void closed subsets of a T_1 space such that...
  18. benorin

    Set of p-adic integers is homeomorphic to Cantor set; how?

    Could somebody explain with due brevity why/how the set of p-adic integers is homeomorphic to the Cantor set less one point for any prime p? This is a quote from Wikipedia:Cantor Set: "The Cantor set is also homeomorphic to the p-adic integers, and, if one point is removed from it, to the...
  19. benorin

    Exploring the Cardinality of Cantor Set and Real Numbers

    So the problem, and my partial solution are in the attached PDF. I would like feedback on my proof of the first statement, if it is technically correct and if it is good. Any ideas as to how I can use/generalize/extend the present proof to proof the second statement, namely that E (the Cantor...
  20. benorin

    Prove Cantor set is measure zero with style

    One of my HW questions asks me to prove that the usual "middle thirds" Cantor set has Lebesgue measure 0. I know two ways, but they lack style... They are (that you may post): #1) The recursive definition of the Cantor set (call it C) removes successively \frac{1}{3} of the unit interval and...
  21. R

    Cantor Expansion Addition algorithm?

    Could someone help me and write an algorithm to add 2 Cantor expansions. The algorithm to get a decimal number to cantor expansion is: procedure decimal-to-cantor(x: positive integer) n := 1 y := x fy is a temporary variable used so that this procedure won't destroy the original value of...
  22. R

    Cantor Expressions: Solving A-F (2,7,19,87,1000,1M)

    What is the cantor expansion of: A. 2 B. 7 C. 19 D. 87 E. 1000 F. 1,000,000 The algorithm to solve these small problems is the most difficult for me. The algorithm that I came up with states: Asub(n) N! + Asub(n-1) (n-1)! +...+ Asub(2)2! + Asub(1)1!, where Asub1 is an integer with...
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