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.