Cantor Set - Construction and Formula of The Ternary Set

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.