Find the domain of the relation ,`R={(x,y) :x,y in Z ,xy=4}`

Answer

To find the domain of the relation \( R = \{(x,y) : x,y \in \mathbb{Z}, xy = 4\} \), we will follow these steps:

### Step 1: Understand the relation
We are given a relation where the product of \( x \) and \( y \) equals 4, and both \( x \) and \( y \) must be integers.

### Step 2: Express \( y \) in terms of \( x \)
From the equation \( xy = 4 \), we can express \( y \) as:
\[ y = \frac{4}{x} \]
Now, we need to find integer values of \( x \) such that \( y \) is also an integer.

### Step 3: Identify integer values of \( x \)
To ensure \( y \) is an integer, \( x \) must be a divisor of 4. The integer divisors of 4 are:
\[ \pm 1, \pm 2, \pm 4 \]

### Step 4: Calculate corresponding \( y \) values
Now we will calculate \( y \) for each divisor of \( x \):
- If \( x = 1 \), then \( y = \frac{4}{1} = 4 \)
- If \( x = -1 \), then \( y = \frac{4}{-1} = -4 \)
- If \( x = 2 \), then \( y = \frac{4}{2} = 2 \)
- If \( x = -2 \), then \( y = \frac{4}{-2} = -2 \)
- If \( x = 4 \), then \( y = \frac{4}{4} = 1 \)
- If \( x = -4 \), then \( y = \frac{4}{-4} = -1 \)

### Step 5: Compile the domain
The values of \( x \) that yield integer \( y \) values are:
\[ 1, -1, 2, -2, 4, -4 \]
Thus, the domain of the relation \( R \) is:
\[ \{1, -1, 2, -2, 4, -4\} \]

### Final Answer
The domain of the relation \( R \) is:
\[ \{1, -1, 2, -2, 4, -4\} \]

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