Let `A=Z`, the set of integers. Let `R_(1)={(m,n)epsilonZxxZ:(m+4n)` is divisible by 5 in `Z}`.
Let `R_(2)={(m,n)epsilonZxxZ:(m+9n)` is divisible by 5 in `Z}`.
Which one of the following is correct?

Options

A. `R_(1)` is a proper subset of `R_(2)`
B. `R_(2)` is a proper subset of `R_(1)`
C. `R_(1)=R_(2)`
D. `R_(1)` is not a symmetric relation on `Z`

Explanation

To solve the problem, we need to analyze the relations \( R_1 \) and \( R_2 \) defined on the set of integers \( A = \mathbb{Z} \).

### Step 1: Define the Relations
- The relation \( R_1 \) is defined as:
\[ R_1 = \{(m, n) \in \mathbb{Z} \times \mathbb{Z} : m + 4n \text{ is divisible by } 5\} \]
- The relation \( R_2 \) is defined as:
\[ R_2 = \{(m, n) \in \mathbb{Z} \times \mathbb{Z} : m + 9n \text{ is divisible by } 5\} \]

### Step 2: Analyze Relation \( R_1 \)
To find the pairs \( (m, n) \) that satisfy \( m + 4n \equiv 0 \mod 5 \):
1. We can rewrite the condition as:
\[ m \equiv -4n \mod 5 \]
2. The values of \( -4n \mod 5 \) can be computed for \( n = 0, 1, 2, 3, 4 \):
- For \( n = 0 \): \( m \equiv 0 \mod 5 \)
- For \( n = 1 \): \( m \equiv 1 \mod 5 \)
- For \( n = 2 \): \( m \equiv 2 \mod 5 \)
- For \( n = 3 \): \( m \equiv 3 \mod 5 \)
- For \( n = 4 \): \( m \equiv 4 \mod 5 \)

Thus, the relation \( R_1 \) can be represented as:
\[ R_1 = \{(m, n) : m \equiv -4n \mod 5\} \]

### Step 3: Analyze Relation \( R_2 \)
Now, we analyze \( R_2 \) with the condition \( m + 9n \equiv 0 \mod 5 \):
1. We can rewrite this condition as:
\[ m \equiv -9n \mod 5 \]
2. Since \( -9 \equiv 1 \mod 5 \), we have:
\[ m \equiv -n \mod 5 \]
3. The values of \( -n \mod 5 \) can also be computed for \( n = 0, 1, 2, 3, 4 \):
- For \( n = 0 \): \( m \equiv 0 \mod 5 \)
- For \( n = 1 \): \( m \equiv 4 \mod 5 \)
- For \( n = 2 \): \( m \equiv 3 \mod 5 \)
- For \( n = 3 \): \( m \equiv 2 \mod 5 \)
- For \( n = 4 \): \( m \equiv 1 \mod 5 \)

Thus, the relation \( R_2 \) can be represented as:
\[ R_2 = \{(m, n) : m \equiv -n \mod 5\} \]

### Step 4: Compare the Relations
Now we compare \( R_1 \) and \( R_2 \):
- From \( R_1 \), we have \( m \equiv -4n \mod 5 \).
- From \( R_2 \), we have \( m \equiv -n \mod 5 \).

To check if \( R_1 \) is equal to \( R_2 \), we need to see if there exists a consistent relationship between \( -4n \) and \( -n \) modulo 5:
- If we set \( n = 0, 1, 2, 3, 4 \) and check the equivalences, we find that both relations yield the same pairs for \( (m, n) \).

### Conclusion
Since the conditions for both relations yield the same pairs, we conclude that:
\[ R_1 = R_2 \]

### Final Answer
The correct statement is \( R_1 = R_2 \).