Composition
Since the Cantor set is defined as the set of points not excluded, the proportion (i.e., measure) of the unit interval remaining can be found by total length removed. This total is the geometric progression
So that the proportion left is 1 – 1 = 0.
This calculation shows that the Cantor set cannot contain any interval of non-zero length. In fact, it may seem surprising that there should be anything left — after all, the sum of the lengths of the removed intervals is equal to the length of the original interval. However, a closer look at the process reveals that there must be something left, since removing the "middle third" of each interval involved removing open sets (sets that do not include their endpoints). So removing the line segment (1/3, 2/3) from the original interval leaves behind the points 1/3 and 2/3. Subsequent steps do not remove these (or other) endpoints, since the intervals removed are always internal to the intervals remaining. So the Cantor set is not empty, and in fact contains an infinite number of points.
It may appear that only the endpoints are left, but that is not the case either. The number 1/4, for example, is in the bottom third, so it is not removed at the first step, and is in the top third of the bottom third, and is in the bottom third of that, and in the top third of that, and so on ad infinitum—alternating between top and bottom thirds. Since it is never in one of the middle thirds, it is never removed, and yet it is also not one of the endpoints of any middle third. The number 3/10 is also in the Cantor set and is not an endpoint.
In the sense of cardinality, most members of the Cantor set are not endpoints of deleted intervals.
Read more about this topic: Cantor Set
Famous quotes containing the word composition:
“When I think of God, when I think of him as existent, and when I believe him to be existent, my idea of him neither increases nor diminishes. But as it is certain there is a great difference betwixt the simple conception of the existence of an object, and the belief of it, and as this difference lies not in the parts or composition of the idea which we conceive; it follows, that it must lie in the manner in which we conceive it.”
—David Hume (1711–1776)
“There is singularly nothing that makes a difference a difference in beginning and in the middle and in ending except that each generation has something different at which they are all looking. By this I mean so simply that anybody knows it that composition is the difference which makes each and all of them then different from other generations and this is what makes everything different otherwise they are all alike and everybody knows it because everybody says it.”
—Gertrude Stein (1874–1946)
“Since body and soul are radically different from one another and belong to different worlds, the destruction of the body cannot mean the destruction of the soul, any more than a musical composition can be destroyed when the instrument is destroyed.”
—Oscar Cullman. Immortality of the Soul or Resurrection of the Dead? The Witness of the New Testament, ch. 1, Epworth Press (1958)