The following relation is defined on the set of real numbers. A R b iff `|a-b|gt0`.

Answer

To analyze the relation defined on the set of real numbers, where \( a R b \) if and only if \( |a - b| > 0 \), we need to 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. This means we need to check if \( a R a \) holds true for any real number \( a \).

- Substitute \( a \) into the relation:
\[ |a - a| > 0 \]
- Simplifying this gives:
\[ |0| > 0 \]
- Since \( 0 \) is not greater than \( 0 \), the condition does not hold.

**Conclusion**: The relation is **not reflexive**.

### Step 2: Check for Symmetry
A relation is symmetric if \( a R b \) implies \( b R a \). We need to check if \( |a - b| > 0 \) leads to \( |b - a| > 0 \).

- Start with the assumption:
\[ |a - b| > 0 \]
- By the properties of absolute values:
\[ |b - a| = |a - b| \]
- Since \( |a - b| > 0 \), it follows that:
\[ |b - a| > 0 \]

**Conclusion**: The relation is **symmetric**.

### Step 3: Check for Transitivity
A relation is transitive if \( a R b \) and \( b R c \) imply \( a R c \). We need to check if \( |a - b| > 0 \) and \( |b - c| > 0 \) lead to \( |a - c| > 0 \).

- Assume:
\[ |a - b| > 0 \quad \text{and} \quad |b - c| > 0 \]
- We cannot directly conclude \( |a - c| > 0 \) from these inequalities. Instead, we can use the triangle inequality:
\[ |a - c| \leq |a - b| + |b - c| \]
- However, knowing that both \( |a - b| \) and \( |b - c| \) are greater than \( 0 \) does not guarantee that \( |a - c| \) is also greater than \( 0 \). For example, if \( a = 1, b = 2, c = 1.5 \), then \( |a - b| = 1 > 0 \) and \( |b - c| = 0.5 > 0 \), but \( |a - c| = |1 - 1.5| = 0.5 > 0 \) does not hold in all cases.

**Conclusion**: The relation is **not transitive**.

### Final Summary
- The relation \( a R b \) defined by \( |a - b| > 0 \) is **not reflexive**, **symmetric**, and **not transitive**.