Construction and Formula of The Ternary Set
The Cantor ternary set is created by repeatedly deleting the open middle thirds of a set of line segments. One starts by deleting the open middle third (1⁄3, 2⁄3) from the interval, leaving two line segments: ∪ . Next, the open middle third of each of these remaining segments is deleted, leaving four line segments: ∪ ∪ ∪ . This process is continued ad infinitum, where the nth set
The Cantor ternary set contains all points in the interval that are not deleted at any step in this infinite process.
The first six steps of this process are illustrated below.
An explicit formula for the Cantor set is
Let us note that this description of the Cantor set does not characterize the complement of the Cantor set exactly, since the sets given by the formula
are not disjoint.
The proof of the formula above is done by the idea of self-similarity transformations and can be found in detail.
Read more about this topic: Cantor Set
Famous quotes containing the words construction, formula and/or set:
“There’s no art
To find the mind’s construction in the face.”
—William Shakespeare (1564–1616)
“For the myth is the foundation of life; it is the timeless schema, the pious formula into which life flows when it reproduces its traits out of the unconscious.”
—Thomas Mann (1875–1955)
“In French literature, you can choose “à la carte”; in Spanish literature, there is only the set meal.”
—José Bergamín (1895–1983)