The task presented in the image involves finding the composition of two relations $R$ and $S$, denoted as $S \circ R$. Let me break it down step-by-step:

Definitions:

1. Relation $R$: A subset of $\{1, 2, 3\} \times \{1, 2, 3, 4\}$, which is defined as: $R = \{(1, 1), (1, 4), (2, 3), (3, 1), (3, 4)\}.$

2. Relation $S$: A subset of $\{1, 2, 3, 4\} \times \{0, 1, 2\}$, which is defined as: $S = \{(1, 0), (2, 0), (3, 1), (3, 2), (4, 2)\}.$

3. Composition of Relations $S \circ R$: The composition $S \circ R$ is a relation that connects an element $a$ from the domain of $R$ to an element $c$ in the range of $S$, such that there exists an intermediate $b$ where: $(a, b) \in R \quad \text{and} \quad (b, c) \in S.$

   In simpler terms, $(a, c) \in S \circ R$ if there exists some $b$ such that $R(a, b)$ and $S(b, c)$.

Step-by-Step Solution:

To compute $S \circ R$, we examine each pair in $R$ and check if the second element of the pair matches the first element of any pair in $S$. If they match, we create a new pair where the first element is from $R$, and the second element is from $S$.

1. Pairs in $R$:

   • $(1, 1)$, $(1, 4)$, $(2, 3)$, $(3, 1)$, $(3, 4)$.

2. Pairs in $S$:

   • $(1, 0)$, $(2, 0)$, $(3, 1)$, $(3, 2)$, $(4, 2)$.

3. Finding Matches:

   • For $(1, 1) \in R$: The second element is $1$. Look for pairs in $S$ where the first element is $1$. The pair $(1, 0) \in S$ matches. Result: $(1, 0)$.
   • For $(1, 4) \in R$: The second element is $4$. Look for pairs in $S$ where the first element is $4$. The pair $(4, 2) \in S$ matches. Result: $(1, 2)$.
   • For $(2, 3) \in R$: The second element is $3$. Look for pairs in $S$ where the first element is $3$. The pairs $(3, 1)$ and $(3, 2) \in S$ match. Results: $(2, 1)$, $(2, 2)$.
   • For $(3, 1) \in R$: The second element is $1$. Look for pairs in $S$ where the first element is $1$. The pair $(1, 0) \in S$ matches. Result: $(3, 0)$.
   • For $(3, 4) \in R$: The second element is $4$. Look for pairs in $S$ where the first element is $4$. The pair $(4, 2) \in S$ matches. Result: $(3, 2)$.

Final Result:

The composition $S \circ R$ is the set of pairs: $S \circ R = \{(1, 0), (1, 2), (2, 1), (2, 2), (3, 0), (3, 2)\}.$

Explanation of the Diagram:

The diagram visually represents the relations:

1. The nodes labeled $1, 2, 3, 4$ and the arrows between them represent the relation $R$.
2. The nodes labeled $0, 1, 2$ and their connections represent the relation $S$.
3. The combined connections (paths from $R$ to $S$ via intermediate elements) represent the composition $S \circ R$.

Do you want further clarification or details about this explanation?

5 Related Questions:

1. What is the formal definition of a relation in set theory?
2. How can the composition of relations be represented using matrices?
3. Is the composition of relations $S \circ R$ always commutative? Why or why not?
4. What are some practical applications of relation composition in computer science?
5. How can graph theory be used to visualize relation compositions effectively?

Tip:

To ensure accuracy when performing relation compositions, always systematically check every pair in $R$ and $S$ for matches, and verify intermediate elements carefully.