In each of the following, state which of the ordered pairs belong to the given relations? `{(x,y):xy=12},(3,4),(4,3),(12,0),(0,12),(12,1),(6,2),(7,5)`

Answer

To determine which of the ordered pairs belong to the relation defined by the condition \( xy = 12 \), we will evaluate each ordered pair one by one.

### Step-by-Step Solution:

1. **Relation Definition**: The relation is defined as \( R = \{(x,y) : xy = 12\} \).

2. **Evaluate Each Ordered Pair**:
- **For (3, 4)**:
- \( x = 3 \) and \( y = 4 \)
- Calculate \( 3 \times 4 = 12 \)
- Since \( 12 = 12 \), the pair \( (3, 4) \) belongs to the relation.

- **For (4, 3)**:
- \( x = 4 \) and \( y = 3 \)
- Calculate \( 4 \times 3 = 12 \)
- Since \( 12 = 12 \), the pair \( (4, 3) \) belongs to the relation.

- **For (12, 0)**:
- \( x = 12 \) and \( y = 0 \)
- Calculate \( 12 \times 0 = 0 \)
- Since \( 0 \neq 12 \), the pair \( (12, 0) \) does not belong to the relation.

- **For (0, 12)**:
- \( x = 0 \) and \( y = 12 \)
- Calculate \( 0 \times 12 = 0 \)
- Since \( 0 \neq 12 \), the pair \( (0, 12) \) does not belong to the relation.

- **For (12, 1)**:
- \( x = 12 \) and \( y = 1 \)
- Calculate \( 12 \times 1 = 12 \)
- Since \( 12 = 12 \), the pair \( (12, 1) \) belongs to the relation.

- **For (6, 2)**:
- \( x = 6 \) and \( y = 2 \)
- Calculate \( 6 \times 2 = 12 \)
- Since \( 12 = 12 \), the pair \( (6, 2) \) belongs to the relation.

- **For (7, 5)**:
- \( x = 7 \) and \( y = 5 \)
- Calculate \( 7 \times 5 = 35 \)
- Since \( 35 \neq 12 \), the pair \( (7, 5) \) does not belong to the relation.

3. **Summary of Results**:
- Ordered pairs that belong to the relation: \( (3, 4), (4, 3), (12, 1), (6, 2) \)
- Ordered pairs that do not belong to the relation: \( (12, 0), (0, 12), (7, 5) \)

### Final Answer:
The ordered pairs that belong to the relation \( \{(x,y):xy=12\} \) are:
- \( (3, 4) \)
- \( (4, 3) \)
- \( (12, 1) \)
- \( (6, 2) \)