In this question, the universal set is the set of all positive integers. Define the set sequences $$ A_{i}, B_{i} $$, and $$ C_{i} $$ as follows: $$ A_{i} $$ is the set of all positive integers that contain up to and including i digits. $$ B_{i} $$ is the set of all positive multiples of 2 that contain up to and including i digits. $$ C_{i} $$ is the set of all positive multiples of 13 that contain up to and including i digits. Check the box(es) for the following statements that are true. (1) $$ C_{2} \cap B_{2}=\{26,52,78\} $$ (2) $$ A_{1} \cap C_{4}=\emptyset $$ (3) $$ A_{2} \cup B_{3} \cup C_{1} $$ is an infinite set. (4) $$ B_{2} \cap A_{5}=B_{5} $$

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Let's evaluate each statement based on the definitions provided:

1. **$$ C_{2} \cap B_{2} = \{26, 52, 78\} $$**  
   - **$$ C_{2} $$**: Multiples of 13 up to 2 digits: $$ \{13, 26, 39, 52, 65, 78, 91\} $$.
   - **$$ B_{2} $$**: Multiples of 2 up to 2 digits: $$ \{2, 4, 6, \ldots, 98\} $$.
   - **Intersection**: The common elements are $$ \{26, 52, 78\} $$.  
   **This statement is true.**

2. **$$ A_{1} \cap C_{4} = \emptyset $$**  
   - **$$ A_{1} $$**: All 1-digit positive integers: $$ \{1, 2, 3, \ldots, 9\} $$.
   - **$$ C_{4} $$**: Multiples of 13 up to 4 digits: $$ \{13, 26, 39, \ldots, 9999\} $$.
   - **Intersection**: There are no 1-digit multiples of 13 since the smallest multiple is 13.  
   **This statement is true.**

3. **$$ A_{2} \cup B_{3} \cup C_{1} $$ is an infinite set.**  
   - **$$ A_{2} $$**: All 2-digit positive integers: $$ \{1, 2, \ldots, 99\} $$.
   - **$$ B_{3} $$**: Multiples of 2 up to 3 digits: $$ \{2, 4, 6, \ldots, 998\} $$.
   - **$$ C_{1} $$**: Multiples of 13 up to 1 digit: $$ \emptyset $$ (since the smallest multiple is 13).
   - **Union**: The union of finite sets $$ A_{2} $$ and $$ B_{3} $$ is still finite (specifically, it contains up to 999 elements).  
   **This statement is false.**

4. **$$ B_{2} \cap A_{5} = B_{5} $$**  
   - **$$ B_{2} $$**: Multiples of 2 up to 2 digits: $$ \{2, 4, \ldots, 98\} $$.
   - **$$ A_{5} $$**: All positive integers up to 5 digits: $$ \{1, 2, \ldots, 99999\} $$.
   - **$$ B_{5} $$**: Multiples of 2 up to 5 digits: $$ \{2, 4, \ldots, 99998\} $$.
   - **Intersection**: $$ B_{2} \cap A_{5} = B_{2} $$, which is a subset of $$ B_{5} $$, but not equal to $$ B_{5} $$.  
   **This statement is false.**

**True Statements:** (1) and (2).
True statements: (1) and (2).

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