Let `A={1,2,3,4}`. Let R be the equivalence relation on `AxxA` defined by `(a,b)R(c,d)` iff `a+d=b+c`. Find `{(1,3)}`.

Answer

To solve the problem, we need to find the equivalence relation for the pair (1, 3) in the context of the relation R defined on the set \( A \times A \) where \( A = \{1, 2, 3, 4\} \). The relation \( R \) is defined such that \( (a, b) R (c, d) \) if and only if \( a + d = b + c \).

### Step-by-Step Solution:

1. **Understanding the Relation**:
We start by understanding the equivalence relation defined on \( A \times A \). The relation states that two pairs \( (a, b) \) and \( (c, d) \) are related if the sum of the first element of the first pair and the second element of the second pair is equal to the sum of the second element of the first pair and the first element of the second pair.

2. **Finding the Relation for (1, 3)**:
We need to find all pairs \( (c, d) \) such that \( (1, 3) R (c, d) \). This means we need to satisfy the equation:
\[ 1 + d = 3 + c \]
Rearranging this gives us:
\[ d = c + 2 \]

3. **Identifying Valid Pairs**:
Since \( c \) and \( d \) must be elements of the set \( A = \{1, 2, 3, 4\} \), we will find valid values for \( c \) and corresponding \( d \):
- If \( c = 1 \), then \( d = 1 + 2 = 3 \) → Valid pair: \( (1, 3) \)
- If \( c = 2 \), then \( d = 2 + 2 = 4 \) → Valid pair: \( (2, 4) \)
- If \( c = 3 \), then \( d = 3 + 2 = 5 \) → Invalid (5 is not in A)
- If \( c = 4 \), then \( d = 4 + 2 = 6 \) → Invalid (6 is not in A)

4. **Conclusion**:
The valid pairs that are equivalent to \( (1, 3) \) under the relation \( R \) are \( (1, 3) \) and \( (2, 4) \). Therefore, the equivalence class of \( (1, 3) \) is:
\[ \{(1, 3), (2, 4)\} \]

### Final Answer:
The equivalence class of \( (1, 3) \) is \( \{(1, 3), (2, 4)\} \).