IV. Partitioning Intervals Using the Algorithm
We will use the partitioning of a set interval according to the algorithm as a method for creating a list of the elements of the interval. Partitioning allows us to construct a list whereby all the elements of the interval will be included in the list.
Below are two examples of partitioning set intervals.
Example 1 -
Let:
S = {1, 2, 3, 4, 5}
IS = [1, 5]
First we note that in defining the interval [1, 5] we have specified the first two elements of the list L. That is L = (1, 5). To continue populating L, partition [1, 5], adding each relative bound to L as it is created, and continue the process until no intervals are left that can be partitioned.
IS = [1, 5] L = (1, 5)
= [1, [3], 5] L = (1, 3, 5)
= [1, 3], [3, 5]
= [1, [2], 3], [3, [4], 5] L = (1, 2, 3, 4, 5)
= [1, 2], [2, 3], [3, 4], [4, 5]
Since no interval in IS can be further subdivided the interval is fully partitioned and L is complete.
Example 2 -
Let:
S = {1, 2, 3, … ω} where ω is the first infinite ordinal number.
IS = [1, ω)
Partition the interval [1, ω) to create a list L of natural numbers.
IS = [1, ω) L = (1)
= [1, [2], ω) L = (1, 2)
= [1, 2], [2, ω)
= [1, 2], [2, [3], ω) L = (1, 2, 3)
= [1, 2], [2, 3], [3, ω)
… …
Taking the procedure to its limit will create a list of all the natural numbers, L = (1, 2, 3, …). Since IS is defined as an upper open interval, ω is not included in L.