Let R be a relation on Z defined by : `R = {(a,b) : a in Z , b in Z, a^(2) = b^(2)}`. Then range of R = _____.

Answer

To find the range of the relation \( R = \{(a, b) : a \in \mathbb{Z}, b \in \mathbb{Z}, a^2 = b^2\} \), we can follow these steps:

### Step 1: Understand the condition \( a^2 = b^2 \)
The equation \( a^2 = b^2 \) implies that the squares of \( a \) and \( b \) are equal. This can happen in two scenarios:
1. \( a = b \)
2. \( a = -b \)

### Step 2: Express \( b \) in terms of \( a \)
From the condition \( a^2 = b^2 \), we can express \( b \) as:
- \( b = a \)
- \( b = -a \)

### Step 3: Determine the possible values of \( b \)
Since \( a \) can take any integer value (as \( a \in \mathbb{Z} \)), for each integer \( a \), \( b \) can be either \( a \) or \( -a \). Therefore, for any integer \( a \), the corresponding values of \( b \) will also be integers.

### Step 4: Conclusion about the range of \( R \)
Since \( b \) can take any integer value (both positive and negative) depending on the value of \( a \), the range of the relation \( R \) is all integers.

Thus, we conclude that:
\[ \text{Range of } R = \mathbb{Z} \]

### Final Answer:
The range of \( R \) is \( \mathbb{Z} \).

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