On R, the set of real numbers define a relation ~ as follows:
`a, b in R a~b` if a-b=0 or irrational Then

Options

A. R is reflexive, symmetric but not transitive
B. R is reflexive, transitive but not symmetri
C. R is anti-symmetric
D. R is an equivalence relation

Answer

To determine the properties of the relation defined on the set of real numbers \( R \) by \( a \sim b \) if \( a - b = 0 \) or \( a - b \) is irrational, we will check if the relation is reflexive, symmetric, and transitive.

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

- Calculate \( a - a \):
\[ a - a = 0 \]
- Since \( 0 = 0 \), it satisfies the condition \( a - b = 0 \).

Thus, the relation is reflexive.

### Step 2: Check Symmetry
A relation is symmetric if \( a \sim b \) implies \( b \sim a \). We need to check if \( a - b = 0 \) or \( a - b \) is irrational implies \( b - a = 0 \) or \( b - a \) is irrational.

- If \( a - b = 0 \):
\[ b - a = 0 \]
This satisfies the condition.

- If \( a - b \) is irrational:
\[ b - a = -(a - b) \]
Since the negative of an irrational number is also irrational, \( b - a \) is irrational.

Thus, the relation is symmetric.

### Step 3: Check Transitivity
A relation is transitive if \( a \sim b \) and \( b \sim c \) implies \( a \sim c \). We need to check if the conditions hold.

- Assume \( a \sim b \) and \( b \sim c \):
- This means \( a - b = 0 \) or \( a - b \) is irrational.
- And \( b - c = 0 \) or \( b - c \) is irrational.

We need to show that \( a - c = 0 \) or \( a - c \) is irrational.

1. **Case 1**: If \( a - b = 0 \) and \( b - c = 0 \):
\[ a - c = (a - b) + (b - c) = 0 + 0 = 0 \]
This satisfies the condition.

2. **Case 2**: If \( a - b \) is irrational and \( b - c \) is irrational:
- Let \( a - b = x \) (irrational) and \( b - c = y \) (irrational).
- Then \( a - c = (a - b) + (b - c) = x + y \).
- The sum of two irrational numbers can be rational or irrational. For example, if \( x = \sqrt{2} \) and \( y = -\sqrt{2} \), then \( x + y = 0 \) (rational).

Thus, \( a - c \) is not guaranteed to be irrational, which means the relation is not transitive.

### Conclusion
The relation is reflexive and symmetric but not transitive. Therefore, it is not an equivalence relation.

### Final Answer
The relation is reflexive and symmetric but not transitive.