Set builder form of the relation
`R={(-2, -7),(-1, -4),(0,-1),(1,2),(2,5)}` is

Options

A. `{(a,b):b=2a-3,a,b, in Z}`
B. `{(x,y):y=3x-1,x,yinZ}`
C. `{(a,b):b=3a-1,a,binN}`
D. `{(u,v):v=3u-1,-2leu lt3andu in Z}`

Answer

To find the set builder form of the relation \( R = \{(-2, -7), (-1, -4), (0, -1), (1, 2), (2, 5)\} \), we will analyze the pairs in the relation and identify a pattern or a rule that describes the relationship between the elements.

### Step 1: Identify the pairs
The relation consists of the following pairs:
- \( (-2, -7) \)
- \( (-1, -4) \)
- \( (0, -1) \)
- \( (1, 2) \)
- \( (2, 5) \)

### Step 2: Analyze the first elements (x-values)
Let's denote the first element of each pair as \( x \):
- For \( x = -2 \), \( y = -7 \)
- For \( x = -1 \), \( y = -4 \)
- For \( x = 0 \), \( y = -1 \)
- For \( x = 1 \), \( y = 2 \)
- For \( x = 2 \), \( y = 5 \)

### Step 3: Find a relationship between x and y
Now, we will look for a mathematical relationship between \( x \) and \( y \).

Calculating the differences:
- When \( x = -2 \), \( y = -7 \): \( y = 3(-2) - 1 = -7 \)
- When \( x = -1 \), \( y = -4 \): \( y = 3(-1) - 1 = -4 \)
- When \( x = 0 \), \( y = -1 \): \( y = 3(0) - 1 = -1 \)
- When \( x = 1 \), \( y = 2 \): \( y = 3(1) - 1 = 2 \)
- When \( x = 2 \), \( y = 5 \): \( y = 3(2) - 1 = 5 \)

From the calculations, we can see that the relationship can be described by the equation:
\[ y = 3x - 1 \]

### Step 4: Define the set builder notation
The set builder notation for the relation can be expressed as:
\[ R = \{ (x, y) | y = 3x - 1, x \in \mathbb{Z} \text{ and } x \text{ takes values } -2, -1, 0, 1, 2 \} \]

### Final Answer
Thus, the set builder form of the relation \( R \) is:
\[ R = \{ (x, y) | y = 3x - 1, x \in \{-2, -1, 0, 1, 2\} \} \]