Let X be the set of all citizens of India. Elements x, y in X are said to be related if the difference of their age is 5 years. Which one of the following is correct ?

Options

A. The relation is an equivalence relation on X.
B. The relation is symmetric but neither reflexive nor transitive
C. The relation is reflexive but neither, symmetric nor transitive
D. None of the above

Explanation

To analyze the relation defined on the set \( X \) of all citizens of India, we need to check whether the relation is reflexive, symmetric, and transitive.

### Step 1: Check for Reflexivity
A relation is reflexive if every element is related to itself. In this case, we need to check if for every citizen \( x \) in \( X \), the difference of their age with themselves is 5 years.

- For any citizen \( x \), the difference of age \( |x - x| = 0 \).
- Since \( 0 \neq 5 \), the relation is **not reflexive**.

### Step 2: Check for Symmetry
A relation is symmetric if for any two elements \( x \) and \( y \) in \( X \), if \( x \) is related to \( y \), then \( y \) is also related to \( x \).

- If \( x \) is related to \( y \), then \( |x - y| = 5 \).
- This implies \( |y - x| = 5 \) as well, since the absolute value function is symmetric.
- Therefore, the relation is **symmetric**.

### Step 3: Check for Transitivity
A relation is transitive if for any three elements \( x \), \( y \), and \( z \) in \( X \), if \( x \) is related to \( y \) and \( y \) is related to \( z \), then \( x \) must be related to \( z \).

- Suppose \( x \) is related to \( y \) (i.e., \( |x - y| = 5 \)) and \( y \) is related to \( z \) (i.e., \( |y - z| = 5 \)).
- This gives us two equations:
1. \( x - y = 5 \) or \( y - x = 5 \)
2. \( y - z = 5 \) or \( z - y = 5 \)

- From these, we can derive:
- If \( x - y = 5 \) and \( y - z = 5 \), then \( x - z = (x - y) + (y - z) = 5 + 5 = 10 \) (not equal to 5).
- If \( y - x = 5 \) and \( z - y = 5 \), then \( z - x = (z - y) + (y - x) = 5 + 5 = 10 \) (not equal to 5).

- In both cases, \( |x - z| \) does not equal 5, hence the relation is **not transitive**.

### Conclusion
The relation defined is:
- **Not reflexive**
- **Symmetric**
- **Not transitive**

Thus, the correct answer is that the relation is symmetric but neither reflexive nor transitive.