To compute $R1 \circ R2$ (the composition of relations $R1$ and $R2$), we follow these steps:

1. The composition $R1 \circ R2$ is defined as the set of all pairs $(a, c)$, where there exists some $b$ such that $(a, b) \in R2$ and $(b, c) \in R1$.

   In other words, for each pair $(a, b) \in R2$, check if there exists a corresponding pair $(b, c) \in R1$. If such a pair exists, then the pair $(a, c)$ is part of the composition $R1 \circ R2$.

Let's go through the steps to calculate $R1 \circ R2$:

• Look at each element $(a, b)$ in $R2$, and check if there's an element $(b, c)$ in $R1$.

Elements of $R2$:

1. $(1, 1)$ — In $R1$, we find pairs $(1, 1)$ and $(1, 5)$, so we add $(1, 1)$ and $(1, 5)$ to $R1 \circ R2$.
2. $(3, 3)$ — There's no corresponding $(3, c)$ in $R1$, so no pair is added.
3. $(3, 5)$ — There's no corresponding $(5, c)$ in $R1$, so no pair is added.
4. $(5, 3)$ — There's no corresponding $(3, c)$ in $R1$, so no pair is added.
5. $(5, 5)$ — In $R1$, we find pairs $(5, 1)$, $(5, 5)$, and $(5, 7)$, so we add $(5, 1)$, $(5, 5)$, and $(5, 7)$ to $R1 \circ R2$.
6. $(7, 7)$ — In $R1$, we find $(7, 5)$ and $(7, 7)$, so we add $(7, 5)$ and $(7, 7)$ to $R1 \circ R2$.

Final result for $R1 \circ R2$:

$R1 \circ R2 = \{ (1, 1), (1, 5), (5, 1), (5, 5), (5, 7), (7, 5), (7, 7) \}$

This is the composition of the two relations.

Would you like more details or clarifications? Here are some related questions to deepen your understanding:

1. How do you interpret the composition of relations geometrically?
2. Can you explain how relations $R1$ and $R2$ are connected to functions?
3. What is the difference between the composition of relations and the union of relations?
4. How would the composition $R2 \circ R1$ differ from $R1 \circ R2$?
5. How would you represent these relations and their compositions using matrices?

Tip: In composition of relations, the order matters! So $R1 \circ R2$ is not necessarily equal to $R2 \circ R1$.