Let us first understand how to approach the problem:

We are given two sets:

• $A = \{2, 3, 5\}$
• $B = \{4, 6, 16\}$

We are also given two relations:

1. $S_1 = \{(a, b) : \frac{a}{b} \}$
2. $S_2 = \{(a, b) : a < b \}$

We need to find the product of the adjacency matrices corresponding to these two relations.

Step 1: Define Adjacency Matrices

Adjacency Matrix for $S_1$

The adjacency matrix $M_{S_1}$ is formed by checking if $\frac{a}{b} \in S_1$, where $a \in A$ and $b \in B$. If the ratio $\frac{a}{b}$ is valid, we assign a 1, otherwise 0.

For each element $a \in A$ and $b \in B$, calculate $\frac{a}{b}$:

\begin{bmatrix} \frac{2}{4} & \frac{2}{6} & \frac{2}{16} \\ \frac{3}{4} & \frac{3}{6} & \frac{3}{16} \\ \frac{5}{4} & \frac{5}{6} & \frac{5}{16} \end{bmatrix}$$ To simplify: $$M_{S_1} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 1 & 0 \end{bmatrix}$$ #### **Adjacency Matrix for $$S_2$$** The adjacency matrix $$M_{S_2}$$ is formed by checking if $$a < b$$, where $$a \in A$$ and $$b \in B$$. If the condition $$a < b$$ holds, assign a 1, otherwise 0. For each element $$a \in A$$ and $$b \in B$$, check if $$a < b$$: $$M_{S_2} = \begin{bmatrix} 2 < 4 & 2 < 6 & 2 < 16 \\ 3 < 4 & 3 < 6 & 3 < 16 \\ 5 < 4 & 5 < 6 & 5 < 16 \end{bmatrix}$$ Simplify: $$M_{S_2} = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 0 & 1 & 1 \end{bmatrix}$$ ### Step 2: Compute the Product of Adjacency Matrices Now, to find the product of the adjacency matrices $$M_{S_1} \times M_{S_2}$$, we perform matrix multiplication. $$M_{S_1} \times M_{S_2} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 1 & 0 \end{bmatrix} \times \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 0 & 1 & 1 \end{bmatrix}$$ Now calculate the product matrix: $$M_{S_1} \times M_{S_2} = \begin{bmatrix} (0 \times 1 + 0 \times 1 + 0 \times 0) & (0 \times 1 + 0 \times 1 + 0 \times 1) & (0 \times 1 + 0 \times 1 + 0 \times 1) \\ (0 \times 1 + 0 \times 1 + 0 \times 0) & (0 \times 1 + 0 \times 1 + 0 \times 1) & (0 \times 1 + 0 \times 1 + 0 \times 1) \\ (1 \times 1 + 1 \times 1 + 0 \times 0) & (1 \times 1 + 1 \times 1 + 0 \times 1) & (1 \times 1 + 1 \times 1 + 0 \times 1) \end{bmatrix}$$ Simplifying: $$M_{S_1} \times M_{S_2} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 2 & 2 & 2 \end{bmatrix}$$ ### Final Result: The product of the adjacency matrices is: $$\boxed{ \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 2 & 2 & 2 \end{bmatrix} }$$ Would you like further details or have any questions? Here are 5 related questions: 1. What is the significance of adjacency matrices in graph theory? 2. How does matrix multiplication represent composition of relations? 3. What happens if the sets $$A$$ and $$B$$ are of different sizes? 4. Can you generalize the method for larger sets $$A$$ and $$B$$? 5. How do we interpret non-binary entries in a product matrix? **Tip:** Always verify if the sets used in relations are well-defined, and check whether operations between them are applicable.