Z is the set of integers. Describe the following relation in set builder form, given its domain and range.
`{(0,-7),(2,-5),(4,-3),(-13,-20),.......}`

Answer

To describe the given relation in set builder form, we will analyze the ordered pairs provided and derive a general rule that defines the relationship between the elements.

### Step-by-Step Solution:

1. **Identify the Ordered Pairs**:
The given ordered pairs are:
\[ \{(0, -7), (2, -5), (4, -3), (-13, -20), \ldots\} \]

2. **Analyze the Relationship**:
We will look for a pattern between the first element (x) and the second element (y) in each ordered pair.
- For the first pair: \(0 - (-7) = 0 + 7 = 7\)
- For the second pair: \(2 - (-5) = 2 + 5 = 7\)
- For the third pair: \(4 - (-3) = 4 + 3 = 7\)
- For the fourth pair: \(-13 - (-20) = -13 + 20 = 7\)

From this analysis, we can see that for each pair, the difference \(x - y = 7\).

3. **Formulate the Relation**:
We can express the relationship as:
\[ x - y = 7 \quad \text{or} \quad y = x - 7 \]

4. **Set Builder Notation**:
Since \(x\) is an integer (as \(Z\) is the set of integers), we can express the relation in set builder notation:
\[ R = \{(x, y) \in Z \times Z \mid y = x - 7\} \]

5. **Identify Domain and Range**:
- **Domain**: The first elements of the pairs are \(0, 2, 4, -13, \ldots\). This suggests that the domain consists of integers that can be expressed in the form \(x\) where \(x\) can take values from the set of integers.
- **Range**: The second elements of the pairs are \(-7, -5, -3, -20, \ldots\). This indicates that the range also consists of integers that can be expressed in the form \(y\) where \(y\) can take values from the set of integers.

### Final Answer:
Thus, the relation can be described in set builder form as:
\[ R = \{(x, y) \in Z \times Z \mid y = x - 7\} \]