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To determine if the relation \( R_1 \) is transitive, we need to check if whenever \( (a, b) \in R_1 \) and \( (b, c) \in R_1 \), it implies that \( (a, c) \in R_1 \).

Given:
- \( A \) is the set of even positive integers less than or equal to 20.
- \( B \) is the set of positive integers less than 20 which are divisible by 6.
- \( R_1 = \{ (a, b) \mid a \in A, b \in B, a \text{ is a factor of } b \} \).

First, identify the sets \( A \) and \( B \):
- \( A = \{ 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 \} \)
- \( B = \{ 6, 12, 18 \} \)

The relation \( R_1 \) contains pairs \((a, b)\) such that \( a \) is a factor of \( b \).

Now, let's list the pairs in \( R_1 \):
- \( (2, 6) \), \( (2, 12) \), \( (2, 18) \)
- \( (6, 6) \), \( (6, 12) \), \( (6, 18) \)
- \( (12, 12) \), \( (12, 18) \)
- \( (18, 18) \)

For \( R_1 \) to be transitive, if \( (a, b) \in R_1 \) and \( (b, c) \in R_1 \), then \( (a, c) \in R_1 \).

Check transitivity:
1. If \( a = 2 \):
   - \( (2, 6) \) and \( (6, 6) \) implies \( (2, 6) \), which is true.
   - \( (2, 6) \) and \( (6, 12) \) implies \( (2, 12) \), which is true.
   - \( (2, 6) \) and \( (6, 18) \) implies \( (2, 18) \), which is true.
   - \( (2, 12) \) and \( (12, 12) \) implies \( (2, 12) \), which is true.
   - \( (2, 12) \) and \( (12, 18) \) implies \( (2, 18) \), which is true.
   - \( (2, 18) \) and \( (18, 18) \) implies \( (2, 18) \), which is true.

2. If \( a = 6 \):
   - \( (6, 6) \) and \( (6, 6) \) implies \( (6, 6) \), which is true.
   - \( (6, 6) \) and \( (6, 12) \) implies \( (6, 12) \), which is true.
   - \( (6, 6) \) and \( (6, 18) \) implies \( (6, 18) \), which is true.
   - \( (6, 12) \) and \( (12, 12) \) implies \( (6, 12) \), which is true.
   - \( (6, 12) \) and \( (12, 18) \) implies \( (6, 18) \), which is true.
   - \( (6, 18) \) and \( (18, 18) \) implies \( (6, 18) \), which is true.

3. If \( a = 12 \):
   - \( (12, 12) \) and \( (12, 12) \) implies \( (12, 12) \), which is true.
   - \( (12, 12) \) and \( (12, 18) \) implies \( (12, 18) \), which is true.
   - \( (12, 18) \) and \( (18, 18) \) implies \( (12, 18) \), which is true.

4. If \( a = 18 \):
   - \( (18, 18) \) and \( (18, 18) \) implies \( (18, 18) \), which is true.

Since all possible combinations satisfy the transitivity condition, the relation \( R_1 \) is transitive.

Would you like any further details or have any questions?

Here are 5 questions you might ask:
1. Is \( R_1 \) reflexive?
2. Is \( R_1 \) symmetric?
3. What are the properties of relation \( R_2 \)?
4. Can you explain the concept of reflexive, symmetric, and transitive relations?
5. How do we determine if a relation is an equivalence relation?

Tip: When analyzing relations, always consider their reflexive, symmetric, and transitive properties to fully understand their behavior.

Explore how to determine if relation R1 is transitive in mathematics using sets A and B defined within the problem. Understand the conditions for transitivity and see detailed examples of checking transitivity for relation R1.

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