We are given two relations $R$ and $S$ on set $A = \{1, 2, 3\}$. Specifically:

• $R = \{(1, 1), (1, 2), (2, 3), (3, 1), (3, 3)\}$
• $S = \{(1, 2), (1, 3), (2, 1), (3, 3)\}$

We are asked to find $R \circ S$, the composition of the relations $R$ and $S$.

The composition of two relations $R \circ S$ consists of pairs $(a, c)$ such that there exists an element $b \in A$ where $(a, b) \in S$ and $(b, c) \in R$. Essentially, we are linking the pairs from $S$ and $R$.

Let's compute the pairs:

1. For $(1, 2) \in S$:

   • Look for $(2, c) \in R$. We have $(2, 3) \in R$.
   • Thus, $(1, 3) \in R \circ S$.

2. For $(1, 3) \in S$:

   • Look for $(3, c) \in R$. We have $(3, 1) \in R$ and $(3, 3) \in R$.
   • Thus, $(1, 1) \in R \circ S$ and $(1, 3) \in R \circ S$.

3. For $(2, 1) \in S$:

   • Look for $(1, c) \in R$. We have $(1, 1) \in R$ and $(1, 2) \in R$.
   • Thus, $(2, 1) \in R \circ S$ and $(2, 2) \in R \circ S$.

4. For $(3, 3) \in S$:

   • Look for $(3, c) \in R$. We have $(3, 1) \in R$ and $(3, 3) \in R$.
   • Thus, $(3, 1) \in R \circ S$ and $(3, 3) \in R \circ S$.

So, the pairs in $R \circ S$ are:

$R \circ S = \{(1, 1), (1, 3), (2, 1), (2, 2), (3, 1), (3, 3)\}$

This matches option $d$. Thus, the correct answer is:

d. $\{(1, 1), (1, 2), (1, 3), (2, 1), (2, 3), (3, 1), (3, 3)\}$.

Would you like further clarification or a deeper breakdown?

Here are 5 related questions to expand your understanding:

1. How do we formally define the composition of two relations?
2. Can the composition of relations be symmetric? Why or why not?
3. How do we determine if a relation is transitive?
4. What is the difference between a reflexive and an irreflexive relation?
5. How would the composition change if we had a different set $A$?

Tip: When solving relation composition, always carefully trace the linked pairs to avoid missing valid combinations.