Further a proof that the cantor ternary set is a locally compact, hausdorff topological group is given. General article fractal dimension and the cantor set. Cantors set and cantors function eric hanson february 5, 2016 contents 1 cantors set 1 1. Vowels in the english alphabet v a, e, i, o, u first seven prime numbers. The cantor set is the set remaining after the sequence of. Analysis 1 colloquium of week 4 cantors setthe set. In an 1875 paper, smith discussed a nowheredense set of positive measure on the real line, and volterra introduced a similar example in 1881. The cantor set was first discovered by h j s smith.
Each point of the cantor set lies at the root of an infinite. With respect to chaos theory, it is a good example of a simple fractal. In particular, the middle thirds cantor set is a uniform set. The purpose of this thesis is to examine certain questions concerning the cantor ternary set. The middle third cantor set, deno ted by c, is a n elegant set in set theory. Refer to the construction of the cantor set on page 6. This set is produced from the unit interval by successively removing. This paper examined signicant properties of the cantor set to illustrate its unique structure. Ternary expansions and the cantor set we now claim that the cantor set consists precisely of numbers of the form 3 x x1 k1 a k 3k. The thirds set is a special case of smithvolterra cantor sets in general, which are addressed in section2. Last month, i wrote about the cantor set, a mathematical space that is an interesting mix of small and large.
While the cantor set is important to set theory, it plays an important role. The cantor set is the set remaining after the sequence of open intervals i n,k2 n. I am trying multiple \put commands, but they just go to the right of one another. Its called the middlethirds cantor set because in general you can construct a class of sets with. We are going to construct a \bizarre nonempty subset of r through an intersection of nested sets. Moreover, from the remarks above it follows that each x 2c has one and only one ternary expansion using only 0s and 2s. Recall that a set x is countable if there is a sequence of.
The cantor ternary set is created by repeatedly deleting the open middle thirds of a set of line segments. Starting with the closed unit interval, the open middle third is taken out, leaving the two closed intervals and. We use the cantorlebesgue function to show there are measurable sets which are not borel. Georg cantor and the battle for transfinite set theory joseph w. The flat parts are the images of all of the middle thirds, and these are all connected by the images of the cantor set itself. We give a heuristic for the number of reduced rationals on cantor s middle thirds set, with a fixed bound on the denominator. Next, from the two remaining closed intervals we remove the open middle third i 2,1 19,29 and i 2,2 79,89, and so on. The cantor set and the cantorlebesgue function note. It is simply a subset of the interval 0,1, but the set has some very interesting properties. What happens if you take a middle fths set instead of a middle thirds set. Numbers in both sets consist of only two distinct numbers from the set f0. Math 2030 fall 2012 6 the cantor middle thirds set in this section we construct the famous cantor middle thirds set k.
The purpose of this article is to discuss exactly how many points there are in the cantor set. This construction has been called the devils staircase since it has infinitely many steps. Math 2030 fall 2012 6 the cantor middlethirds set in this section we construct the famous cantor middlethirds set k. The set c3 consists of those points in 0,1 which are never. In this paper, we study the prototype of fractal of the classical cantor middle third set which consists of points along a line segment, and possesses a number of fascinating properties. Yes, at least the endpoints of the deleted middle third subintervals. A typical treatment of the subject of thirds sets and smithvolterracantor sets is given by vallin. The remaining set is a union of two closed intervals 0,1 3 and 2 3,1 from which we then remove the two open middle thirds 1 9,2 9 and 7 9,8 9. Analysis 1 colloquium of week 4 cantors setthe set without the middle thirds jacob shapiro october 8, 2014 abstract we present the cantor set, its ternary representation, and show how this proves that it has the same cardinality as r.
The sum of two cantor sets wolfram demonstrations project. Ternary expansions and the cantor set geometric series a. Diagram of a variation of the middle third cantor set. The cantor set is an interesting mathematical construction which challenges our intuition about subsets in r. The cantor set is the limit or intersection of all such sets. Begin with the closed real interval 0, 1 and divide it into three equal open subintervals. We give a heuristic for the number of reduced rationals on cantors middle thirds set, with a fixed bound on the denominator. Is there anything left after removing all these middle thirds.
The middle thirds cantor set is obtained by setting n 3 and d 0, 2. C is a closed set with lebesgue measure strictly greater than 17. The construction of the 1 3 cantor set a famous example of a set with nonintegral dimension is the middle third cantor set. We now present the usual middlethird definition of the cantor set. The set of numbers that will never be removed is called the cantor set and it has some amazing properties. This process of division and removal is repeated c \1 n1 c n, where c n is the set that has the middle 3rd interval removed, from each of the intervals from c n 1.
While the cantor set is important to set theory, it plays an important role in other areas of mathematics, such as fractals. Haar measure on the cantor ternary set unt digital library. The cantor set is selfsimilar, consisting of n 2 congruent subsets, each when magni. The smithvolterracantor set is topologically equivalent to the middlethirds cantor set. The size of the cantor set washington university math circle. These objects are sometimes called elements or members of the set. We continue in this way, with e k obtained by deleting the middle third of each interval in e k. R, then dividing it into three intervals of equal length and removing the middle interval. Then the set c2 is obtained by removing from c1 the open intervals 1 9, 2 9 and 7 9, 8 9. I am trying to draw a cantor set in latex and i am having issues doing this. Another way to describe this is to say that c is the set of points in 0.
The smithvolterra cantor set is topologically equivalent to the middle thirds cantor set. A different 2d analogue of the cantor set is the sierpinski carpet, where a square is divided up into nine smaller squares, and the middle one removed. Lebesgue integration on r university of washington. Then the middle thirds of those two intervals are taken out, leaving four intervals of length, and so on. It also lends itself to some interesting counterexamples in measure theory, among other things.
We also describe extensive numerical computations supporting this heuristic. Construct a variant of the cantor set that has positive measure. The cantor set is uncountable trinity college dublin. If the cantor set is homeomorphic to another subset of r, does. In this lecture we construct selfsimilar sets of fractional dimension. This construction has been called the devils staircase since it. Why is the middle third cantor set written as this. Recall c is obtained from the closed interval 0,1 by. The cantor set is an interesting example of an uncountable set of measure zero and has many interesting properties and consequences in the fields of set theory, topology, and fractal theory. The thirds set is a special case of smithvolterracantor sets in general, which are addressed in section2. The cantor set as we know it today followed in 1883. Analysis 1 colloquium of week 4 cantors setthe set without.
Pdf the cantor set is an interesting example of an uncountable set of measure zero and has many interesting. Fractal geometry mathematical foundations and applications. Delete the open middle third of the segment, leaving behind two closed segments. Math 270 spring 2003 homework 8 due friday april 11, 2003. The remaining squares are then further divided into nine each and the middle removed, and so on ad infinitum. We also explore the hausdorff dimension of the intersection of two middle third cantor sets as the sets are. For example, there are infinitely many numbers in the cantor set even uncountably many numbers, but it contains no intervals of numbers and its total length is zero. We can think of the cantor set as what is left of c.
Begin with the closed real interval 0,1 and divide it into three equal open subintervals. The distribution of rational numbers on cantors middle. If the cantor set is homeomorphic to another subset of r, does that set also have measure 0. The middle third cantor set f consists of the numbers that are in e. At each step, we delete the open middle third subinterval of each interval obtained in the previous step. The cantor set and the cantor function tma4225 foundations of analysis clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. In this paper, we study the prototype of fractal of the classical cantor middlethird set which consists of points along a line segment, and possesses a number of fascinating properties. We iteratively remove the open middle onethird of closed subintervals of ias follows. The cantor ternary set is a subset of the real interval s0 0,1. Cantors set is the set c left after this procedure of deleting the open middle third subinterval is performed in nitely many times. Real analysis questions october 2012 contents 1 measure theory 2 2 riemann integration 3. The second chapter deals with proving that the cantor ternary set is equivalent to the middle thirds set of 0,1, closed, compact, and has lebesgue measure zero.
Cantor set and cantorlebesgue function 1 section 2. Usually, it is used to construct so me counterexamples in analysis. Next, from the two remaining closed intervals we remove the open middle third. Cantor set and its properties zhixing guo university of california, santa barbara april 23, 2014. A typical treatment of the subject of thirds sets and smithvolterra cantor sets is given by vallin. To form c1, we remove the middle third of the interval.
467 158 79 538 308 677 981 579 154 482 740 745 653 126 651 1284 1219 1249 233 881 1111 52 618 1371 861 730 687 965 861 1346 125 501 1486 139 1120 141 363