Question

Among the following four statements about countability and uncountability of different sets, which is the correct statement?

• A. The set \( \bigcup \left\{ x \in \mathbb{R} : x = \sum_{i=0}^{n} 10^i a_i, { where } a_i \in \{1, 2\} { for } i = 0, 1, 2, \dots, n \right\} \) is uncountable.
• B. The set \( \left\{ x \in (0,1) : x = \sum_{n=1}^{\infty} \frac{a_n}{10^n}, { where } a_n = 1 { or } 2 { for each } n \in \mathbb{N} \right\} \) is uncountable.
• C. There exists an uncountable set whose elements are pairwise disjoint open intervals in \( \mathbb{R} \).
• D. The set of all intervals with rational end points is uncountable.

Hint:

For problems involving uncountability, look for representations of sets that involve infinite sequences or structures with binary-like expansions, as these often indicate uncountable sets.

Answer 1

The Correct Option is B

Step 1: Understanding the representation of the set in option (B).
The set in option (B) consists of numbers in the interval \( (0,1) \) that can be represented as an infinite series:

\[ x = \sum_{n=1}^{\infty} \frac{a_n}{10^n} \]
where each \( a_n \) can be either 1 or 2 for all \( n \in \mathbb{N} \). This is a form of decimal expansion where the digits of the number are restricted to only two values: 1 and 2.

Step 2: Recognizing the connection to binary sequences.
The decimal expansion described in option (B) is essentially a binary-like expansion, except the digits are not restricted to 0 and 1 but are allowed to be 1 or 2. Specifically, every \( a_n \in \{1, 2\} \) can be mapped to a binary sequence as follows:
- \( a_n = 1 \) maps to 0
- \( a_n = 2 \) maps to 1

Therefore, the set described in option (B) corresponds to all possible binary sequences, where each sequence is infinite and consists of either 0s or 1s.

Step 3: Conclusion about uncountability.
The set of all infinite binary sequences is known to be uncountable, as it has the same cardinality as the real numbers in the interval \( [0, 1] \). This is a direct application of Cantor's diagonal argument, which shows that the set of all infinite binary sequences is uncountable.

Thus, the set described in option (B) is uncountable.

Step 4: Verifying other options.
Option (A): The set described in option (A) is a union of sets of real numbers with finite decimal expansions. Since each set consists of a finite number of elements for each \( n \), the union of these sets is countable, not uncountable.
Option (C): Although there exist uncountable sets of disjoint open intervals, this specific statement is more complex and does not directly relate to the uncountability of the set described in option (B).
Option (D): The set of intervals with rational endpoints is countable, since the rationals are countable and each interval is uniquely defined by two rational endpoints.

Step 5: Final conclusion.
Hence, the correct answer is option (B).