Quiz: The Cantor Set — 8 questions

1. How is the Cantor set $F$ defined in the middle-third construction?

A. As the union of all removed middle thirds

B. As the intersection of all surviving sets $F_n$

C. As the collection of interval endpoints at every stage

D. As the first surviving set after one removal

Answer: (B)

Explanation

The Cantor set is formed by intersecting the nested sequence of surviving sets $F_n$. A set such as $F_n$ represents a finite stage, not the completed Cantor set itself.

2. Which ternary-expansion property characterizes membership in the Cantor set?

A. The expansion contains the digit 1 at every position

B. The expansion terminates after its first nonzero digit

C. The expansion uses only the digits 1 and 2

D. The expansion uses only the digits 0 and 2

Answer: (D)

Explanation

A real number in the unit interval belongs to the Cantor set when it has a ternary expansion using only 0 and 2. A ternary digit 1 signals that the number lies in a removed middle third.

3. What is removed from each surviving interval at every stage of the Cantor construction?

A. Its midpoint together with both endpoints

B. Its closed middle third

C. Its open middle third

D. Its two outer thirds

Answer: (C)

Explanation

Each step removes the open middle third from every surviving closed interval. The endpoints of that middle third remain because the removed interval is open.

4. If $F_n$ is the union of $2^n$ intervals $[k/3^n,(k+1)/3^n]$, how is $F_{n+1}$ formed?

A. By removing the endpoints from every interval in $F_n$

B. By joining each adjacent pair of intervals in $F_n$

C. By removing the open middle third of every interval in $F_n$

D. By removing both closed outer thirds from every interval in $F_n$

Answer: (C)

Explanation

The next stage is obtained by deleting the open middle third from each interval present in $F_n$. The endpoints remain, so the surviving pieces are closed intervals rather than open ones.

5. What is the defining characteristic of the Cantor set in terms of ternary expansions?

A. It consists exactly of the real numbers in the unit interval whose ternary expansions use only the digits 0 and 2.

B. It contains only the rational numbers within the interval [0,1].

C. It is composed of all real numbers in the interval [0,1] with ternary expansions that include the digit 1.

D. It includes all real numbers in the unit interval with finite ternary expansions.

Answer: (A)

Explanation

The Cantor set is characterized by containing exactly those points in the unit interval whose ternary expansions use only the digits 0 and 2. Numbers with a 1 in their ternary expansion are removed during the construction process, so they are not part of the Cantor set.

6. What is the cardinality of the Cantor set in relation to the unit interval?

A. It has exactly countably infinite points.

B. It has as many points as the unit interval.

C. It has fewer points than the set of natural numbers.

D. It has a finite number of points.

Answer: (B)

Explanation

The Cantor set has a non-denumerable number of points, meaning it has as many points as the entire unit interval. This is despite its geometric thinness, as it can be put into a one-to-one correspondence with the points in the interval.

7. What is the primary purpose of the ternary characterization of the Cantor set?

A. To prove that the set contains only rational numbers.

B. To show that the set contains all numbers with ternary expansions including 1.

C. To demonstrate that the set is dense in the unit interval.

D. To identify the set as consisting of numbers with ternary expansions using only 0 and 2.

Answer: (D)

Explanation

The ternary characterization specifies that the Cantor set consists exactly of numbers in the unit interval whose ternary expansions use only the digits 0 and 2. This distinguishes members of the set based on their expansion pattern, with digit 1 indicating a removed middle third.

8. How does the ternary characterization define the members of the Cantor set?

A. They are the real numbers in the unit interval whose ternary expansions use only the digits 0 and 2.

B. They are the real numbers in the unit interval with finite ternary expansions.

C. They are the real numbers in the unit interval whose ternary expansions use only the digits 1 and 2.

D. They are the real numbers in the unit interval whose ternary expansions include the digit 1.

Answer: (A)

Explanation

The Cantor set consists exactly of the real numbers in the unit interval whose ternary expansions use only the digits 0 and 2. This excludes numbers with the digit 1, which indicates a removed middle third.