`R_(1)` on Z defined by `(a,b)inR_(1) " iff "|a-b|le7, R_(2)` on Q defined by `(a,b)inR_(2) " iff "ab=4 and R_(3)` on R defined by `(a, b)inR_(3)" iff "a^(2)-4ab+3ab^(2)=0`
Relation `R_(2)` is

Options

A. reflexive
B. symmetric
C. transitive
D. equivalence

Explanation

To analyze the relation \( R_2 \) defined on \( \mathbb{Q} \) such that \( (a, b) \in R_2 \) if and only if \( ab = 4 \), we will check if this relation is reflexive, symmetric, and transitive.

### Step 1: Check for Reflexivity
A relation is reflexive if every element is related to itself. For \( R_2 \), we need to check if \( (a, a) \in R_2 \) for all \( a \in \mathbb{Q} \).

- For \( (a, a) \) to be in \( R_2 \), we need \( a \cdot a = 4 \), which simplifies to \( a^2 = 4 \).
- The solutions to this equation are \( a = 2 \) and \( a = -2 \).
- Therefore, \( (2, 2) \) and \( (-2, -2) \) are in \( R_2 \), but elements like \( (1, 1) \) or \( (3, 3) \) are not in \( R_2 \).

**Conclusion**: \( R_2 \) is not reflexive since not all elements \( a \in \mathbb{Q} \) satisfy \( a^2 = 4 \).

### Step 2: Check for Symmetry
A relation is symmetric if whenever \( (a, b) \in R_2 \), then \( (b, a) \in R_2 \).

- Assume \( (a, b) \in R_2 \). This means \( ab = 4 \).
- We can rearrange this to \( ba = 4 \), which implies \( (b, a) \in R_2 \).

**Conclusion**: \( R_2 \) is symmetric.

### Step 3: Check for Transitivity
A relation is transitive if whenever \( (a, b) \in R_2 \) and \( (b, c) \in R_2 \), then \( (a, c) \in R_2 \).

- Assume \( (a, b) \in R_2 \) and \( (b, c) \in R_2 \). This means \( ab = 4 \) and \( bc = 4 \).
- From \( ab = 4 \), we can express \( b = \frac{4}{a} \).
- Substituting \( b \) in \( bc = 4 \) gives us \( \frac{4}{a}c = 4 \), which simplifies to \( c = a \).
- However, for \( (a, c) \) to be in \( R_2 \), we need \( ac = 4 \). This is not guaranteed since \( a \) and \( c \) can be different values.

**Conclusion**: \( R_2 \) is not transitive.

### Final Conclusion
Since \( R_2 \) is not reflexive and not transitive, but it is symmetric, we conclude that \( R_2 \) is a symmetric relation but not an equivalence relation.