Let R be relation from a set A to a set B, then (1)`R= AuuB` (2)`R=AnnB` (3)`RsubeA`x`B` (4)`RsubeB`x`A`

Options

A. `R sub B xx A`
B. `R sub A xx B`
C. `R = A xx B`
D. `A xx B sub R`

Answer

To solve the question, we need to analyze the definitions of the relations given in the options and determine which one correctly describes a relation \( R \) from set \( A \) to set \( B \).

### Step-by-Step Solution:

1. **Understanding Relations**: A relation \( R \) from set \( A \) to set \( B \) is defined as a subset of the Cartesian product \( A \times B \). This means that \( R \) consists of ordered pairs where the first element comes from set \( A \) and the second element comes from set \( B \).

2. **Analyzing the Options**:
- **Option (1)**: \( R = A \cup B \)
This option suggests that \( R \) is the union of sets \( A \) and \( B \). However, a relation from \( A \) to \( B \) cannot be simply the union of these sets. Thus, this option is incorrect.

- **Option (2)**: \( R = A \cap B \)
This option suggests that \( R \) is the intersection of sets \( A \) and \( B \). Similar to the first option, a relation cannot be defined as the intersection of two sets unless they have common elements that can form pairs. Therefore, this option is also incorrect.

- **Option (3)**: \( R \subseteq A \times B \)
This option states that \( R \) is a subset of the Cartesian product \( A \times B \). This is a correct definition of a relation from \( A \) to \( B \) since it allows for any subset of ordered pairs formed by elements of \( A \) and \( B \).

- **Option (4)**: \( R \subseteq B \times A \)
This option suggests that \( R \) is a subset of the Cartesian product \( B \times A \). This would imply that the first element of each ordered pair comes from \( B \) and the second from \( A \), which does not conform to the definition of a relation from \( A \) to \( B \). Thus, this option is incorrect.

3. **Conclusion**: Based on the analysis, the only correct option that describes a relation from set \( A \) to set \( B \) is:
\[ R \subseteq A \times B \]

### Final Answer:
The correct answer is **Option (3)**: \( R \subseteq A \times B \).