S={3,7, 8}.Relation R is defined on set S,where R ={(x,y)Ix is prime or y is prime} write adjecency matrix

To solve this problem, we need to create the adjacency matrix for the relation \( R \) defined on set \( S = \{3, 7, 8\} \). The relation \( R \) includes pairs \((x, y)\) such that either \( x \) or \( y \) is a prime number.

### Step 1: Identify Prime Numbers in Set \( S \)
- The set \( S = \{3, 7, 8\} \) contains the following prime numbers: \( 3 \) and \( 7 \).
- The number \( 8 \) is not prime.

### Step 2: Determine the Relation \( R \)
The relation \( R \) is defined by the condition that at least one of the elements in the pair \((x, y)\) is prime.

We check each possible pair \((x, y)\) where \( x, y \in S \):
- \( (3, 3) \): Both \( 3 \) and \( 3 \) are prime. Thus, \( (3, 3) \in R \).
- \( (3, 7) \): \( 3 \) is prime. Thus, \( (3, 7) \in R \).
- \( (3, 8) \): \( 3 \) is prime. Thus, \( (3, 8) \in R \).
- \( (7, 3) \): \( 7 \) is prime. Thus, \( (7, 3) \in R \).
- \( (7, 7) \): Both \( 7 \) and \( 7 \) are prime. Thus, \( (7, 7) \in R \).
- \( (7, 8) \): \( 7 \) is prime. Thus, \( (7, 8) \in R \).
- \( (8, 3) \): \( 3 \) is prime. Thus, \( (8, 3) \in R \).
- \( (8, 7) \): \( 7 \) is prime. Thus, \( (8, 7) \in R \).
- \( (8, 8) \): Neither \( 8 \) nor \( 8 \) is prime. Thus, \( (8, 8) \notin R \).

### Step 3: Construct the Adjacency Matrix
The adjacency matrix is a square matrix that represents the relation \( R \) on \( S \). The matrix will have dimensions \( 3 \times 3 \) since \( |S| = 3 \). 

The matrix \( A \) is defined as:
- \( A[i][j] = 1 \) if \( (S_i, S_j) \in R \),
- \( A[i][j] = 0 \) otherwise.

Let's assign the elements of \( S \) as follows:
- \( S_1 = 3 \)
- \( S_2 = 7 \)
- \( S_3 = 8 \)

Now, we populate the matrix \( A \):

\[
A = \begin{pmatrix}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 0
\end{pmatrix}
\]

### Explanation:
- The first row corresponds to element 3, and since 3 is prime, it connects with all elements, so all entries in the first row are 1.
- The second row corresponds to element 7, and since 7 is prime, it also connects with all elements, so all entries in the second row are 1.
- The third row corresponds to element 8, which is not prime, so it only connects with the primes 3 and 7, making the third row \( (1, 1, 0) \).

### Final Answer:
The adjacency matrix for the relation \( R \) is:

\[
\begin{pmatrix}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 0
\end{pmatrix}
\]

Would you like to go over any specific part in more detail?

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**Relative Questions:**
1. What is an adjacency matrix and how is it used in graph theory?
2. How do you determine if a number is prime?
3. What would the adjacency matrix look like if all elements in \( S \) were prime?
4. How does the adjacency matrix change if the relation \( R \) is defined differently?
5. Can you explain how to represent directed and undirected graphs using adjacency matrices?

**Tip:** The adjacency matrix is particularly useful for understanding the structure of a graph or relation, as it gives a clear visual of which elements are connected.

Learn how to construct the adjacency matrix for relation R defined on set S={3, 7, 8}, where the relation includes pairs (x, y) such that either x or y is prime. This problem explores concepts in number theory and graph theory.

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