Let A = {2,3, 5}, B = {4, 6, 16} and S1 = {(a, b): a/b}, S2 = {(a, b): a<b; be two relation from set A
to set B. Find the product of the adjacency matrices Ms1 and Ms2

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 \( \frac{a}{b} \):

\[
M_{S_1} =
\begin{bmatrix}
\frac{2}{4} & \frac{2}{6} & \frac{2}{16} \\
\frac{3}{4} & \frac{3}{6} & \frac{3}{16} \\
\frac{5}{4} & \frac{5}{6} & \frac{5}{16}
\end{bmatrix}
\]

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.

Let A = {2,3,5}, B = {4,6,16} and S1 = {(a, b): a/b}, S2 = {(a, b): a<b; be two relation from set A to set B. Find the product of the adjacency matrices Ms1 and Ms2.

Explore how to find the product of adjacency matrices for relations between sets A = {2, 3, 5} and B = {4, 6, 16}, including matrix multiplication, adjacency matrix formation, and relational properties. Suitable for students in Grades 11-12.

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