If the relation R in a set of integers defined as R = {(a, b): (a + b) is divisible by 7}, is an equivalence relation, then the equivalence class containing 0 is:

Options

A. {0, 7, 14, 21, ...}
B. {1, 2, 3, 4, 5, ..... }
C. R
D. {....-21, -14, -7, 0, 7, 14, ...}

Answer

To solve the problem, we need to determine the equivalence class containing 0 for the relation \( R = \{(a, b) : (a + b) \text{ is divisible by } 7\} \) in the set of integers.

### Step-by-Step Solution:

1. **Understanding the Relation**:
The relation \( R \) states that two integers \( a \) and \( b \) are related if the sum \( a + b \) is divisible by 7. This means there exists an integer \( k \) such that:
\[ a + b = 7k \]

2. **Finding the Equivalence Class of 0**:
We want to find the equivalence class containing 0, denoted as \( [0] \). This class consists of all integers \( b \) such that:
\[ 0 + b \text{ is divisible by } 7 \]
This simplifies to:
\[ b \text{ is divisible by } 7 \]

3. **Identifying the Multiples of 7**:
The integers that are divisible by 7 can be expressed in the form:
\[ b = 7k \quad \text{for } k \in \mathbb{Z} \]
This means \( b \) can take values like \( \ldots, -21, -14, -7, 0, 7, 14, 21, \ldots \).

4. **Conclusion**:
Therefore, the equivalence class containing 0 is:
\[ [0] = \{ \ldots, -21, -14, -7, 0, 7, 14, 21, \ldots \} \]
This corresponds to option 4 in the given choices.

### Final Answer:
The equivalence class containing 0 is:
\[ \{ \ldots, -21, -14, -7, 0, 7, 14, 21, \ldots \} \]