
Concept explainers
(a)
To Find: The example of the some numbers in the cantor set.
(a)

Answer to Problem 65E
The example of the some numbers in the cantor set is obtained.
Explanation of Solution
Given:
The total length of all the intervals that are removed is 1
The closed interval
The two interval is left as
Calculation:
To measure the size of the cantor set sue the infinite series by removing the middle terms 0, 1 and segment size is,
The first term is
Then,
Then, the series is,
Consider for the second part 0 and 1 are not removed and the other numbers still in the cantor are
This, implies that
(b)
To Prove: The sum of the areas of the removed squares is 1 and this shows that implies that the sierpinski carpet has the area 0.
(b)

Explanation of Solution
Given:
The centre of the square of side 1 then removing the centres of the eight smaller squares and so on.
The given diagram is shown in Figure 1
Figure 1
The sum of the areas of the removed square is 1
Calculation:
Consider the area of the square is,
Then, the area is,
Hence, proved.
Chapter 8 Solutions
Single Variable Calculus: Concepts and Contexts, Enhanced Edition
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