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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.

R dan S memiliki relasi pada A = {1, 2, 3}: R = {(1, 1), (1, 2), (2, 3), (3, 1), (3, 3)}, S= {(1, 2), (1, 3), (2, 1), (3, 3)}, maka R o S adalah?

Learn how to compute the composition of relations for the set A = {1, 2, 3}, where R = {(1, 1), (1, 2), (2, 3), (3, 1), (3, 3)} and S = {(1, 2), (1, 3), (2, 1), (3, 3)}. This example explores the composition of relations and the necessary steps to find R o S.

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