If R is a relation on `NxxN` defined by (a,b) R (c,d) iff a+d=b+c, then

Options

A. reflexive
B. symmetric
C. transitive
D. all of these

Answer

To solve the problem, we need to analyze the relation \( R \) defined on \( \mathbb{N} \times \mathbb{N} \) by the condition \( (a,b) R (c,d) \) if and only if \( a + d = b + c \). We will check if this relation is reflexive, symmetric, and transitive.

### Step 1: Check for Reflexivity
A relation \( R \) is reflexive if every element is related to itself. For our relation, we need to check if \( (a, b) R (a, b) \).

- We have \( a + b = b + a \) (which is true due to the commutative property of addition).
- Therefore, \( (a, b) R (a, b) \) holds for all \( (a, b) \).

**Conclusion**: The relation is reflexive.

### Step 2: Check for Symmetry
A relation \( R \) is symmetric if whenever \( (a, b) R (c, d) \), then \( (c, d) R (a, b) \) also holds.

- Assume \( (a, b) R (c, d) \), which means \( a + d = b + c \).
- We need to show that \( (c, d) R (a, b) \), which means \( c + b = d + a \).
- From \( a + d = b + c \), we can rearrange it to \( c + b = d + a \) (by simply switching sides).

**Conclusion**: The relation is symmetric.

### Step 3: Check for Transitivity
A relation \( R \) is transitive if whenever \( (a, b) R (c, d) \) and \( (c, d) R (x, y) \), then \( (a, b) R (x, y) \) must also hold.

- Assume \( (a, b) R (c, d) \) and \( (c, d) R (x, y) \).
- This means:
1. \( a + d = b + c \) (1)
2. \( c + y = d + x \) (2)

- We need to show that \( (a, b) R (x, y) \), which means \( a + y = b + x \).
- From (1), we can express \( d \) in terms of \( a, b, c \): \( d = b + c - a \).
- Substitute \( d \) in (2):
\( c + y = (b + c - a) + x \)
Simplifying gives:
\( c + y = b + c - a + x \)
Cancel \( c \) from both sides:
\( y = b - a + x \)
Rearranging gives:
\( a + y = b + x \).

**Conclusion**: The relation is transitive.

### Final Conclusion
Since the relation \( R \) is reflexive, symmetric, and transitive, it is an equivalence relation.

### Answer
The relation \( R \) is an equivalence relation.

---