Which one of the following is not an equivalence relation?

Options

A. `R_(1)` on Z defined by a `R_(1) b iff a - b` is divisible by m, where m is a fixed positive integer.
B. `R_(2)` on R defined by a `R_(2) b iff 1 + ab gt 0 " for all "a, b in R`.
C. `R_(3)` on `N xx N` defined by (a, b) `R_(3)` (c, d) `iff` ad = bc for all a, b, c, d `in` N.
D. `R_(4)` on Z defined by `aR_(4)` b `iff` a-b is an even integer for all `a, b in Z`.

Explanation

To determine which of the given relations is not an equivalence relation, we need to check each relation against the three properties of equivalence relations: reflexivity, symmetry, and transitivity.

### Step-by-Step Solution

1. **Identify the Relations**: We have four relations to evaluate (let's denote them as R1, R2, R3, and R4).

2. **Check R1**:
- **Relation**: A - B is divisible by M.
- **Reflexive**: For any element A, A - A = 0, which is divisible by M. (True)
- **Symmetric**: If A - B is divisible by M, then B - A is also divisible by M. (True)
- **Transitive**: If A - B is divisible by M and B - C is divisible by M, then A - C is also divisible by M. (True)
- **Conclusion**: R1 is an equivalence relation.

3. **Check R2**:
- **Relation**: 1 + aR2b if 1 + ab > 0.
- **Reflexive**: For any a, 1 + a^2 > 0 (True).
- **Symmetric**: If 1 + ab > 0, then 1 + ba > 0 (True).
- **Transitive**: Check with specific values:
- Let a = 1, b = 1/2, c = -1.
- 1 + 1*(1/2) > 0 (True), and 1 + (1/2)(-1) > 0 (False).
- Therefore, 1 + 1*(-1) = 0, which is not greater than 0. (False)
- **Conclusion**: R2 is not transitive, hence not an equivalence relation.

4. **Check R3**:
- **Relation**: If (a, b) is related to (c, d), then ad = bc.
- **Reflexive**: For any a, a*a = a^2 (True).
- **Symmetric**: If ad = bc, then bc = ad (True).
- **Transitive**: If ad = bc and be = cd, then ae = bd (True).
- **Conclusion**: R3 is an equivalence relation.

5. **Check R4**:
- **Relation**: A - B is an even integer.
- **Reflexive**: A - A = 0, which is even (True).
- **Symmetric**: If A - B is even, then B - A is also even (True).
- **Transitive**: If A - B and B - C are even, then A - C is also even (True).
- **Conclusion**: R4 is an equivalence relation.

### Final Conclusion
After checking all relations, we find that R2 is not an equivalence relation because it fails the transitive property.