Let W= set of all persons living in Warangl. Define R on Was follows: `a,b in W, aRb` if the difference between their heights is 2 cm. Then

Options

A. R is reflexive only
B. R is symmetric only
C. R is symmetric and transitive only
D. R is an equivalence relation

Answer

To determine the properties of the relation \( R \) defined on the set \( W \) of all persons living in Warangal, we need to analyze the relation based on the definitions of reflexivity, symmetry, and transitivity.

### Step 1: Understanding the Relation
The relation \( R \) is defined such that for any two persons \( a \) and \( b \) in \( W \), \( aRb \) if the absolute difference between their heights is 2 cm. Mathematically, this can be expressed as:
\[ |h_a - h_b| = 2 \]
where \( h_a \) and \( h_b \) are the heights of persons \( a \) and \( b \) respectively.

### Step 2: Checking Reflexivity
A relation is reflexive if every element is related to itself. For \( R \) to be reflexive, we need:
\[ aRa \quad \text{for all } a \in W \]
This means:
\[ |h_a - h_a| = 2 \]
However, \( |h_a - h_a| = 0 \), which does not equal 2. Therefore, the relation \( R \) is **not reflexive**.

### Step 3: Checking Symmetry
A relation is symmetric if whenever \( aRb \), then \( bRa \). For our relation:
- If \( aRb \), then \( |h_a - h_b| = 2 \).
- This implies \( |h_b - h_a| = 2 \) as well, since absolute value is symmetric.

Thus, if \( aRb \) holds, then \( bRa \) also holds. Therefore, the relation \( R \) is **symmetric**.

### Step 4: Checking Transitivity
A relation is transitive if whenever \( aRb \) and \( bRc \), then \( aRc \) must also hold. For our relation:
- Assume \( aRb \) means \( |h_a - h_b| = 2 \) and \( bRc \) means \( |h_b - h_c| = 2 \).
- This means:
- \( h_a = h_b + 2 \) or \( h_a = h_b - 2 \)
- \( h_b = h_c + 2 \) or \( h_b = h_c - 2 \)

Now, let’s consider cases:
1. If \( h_a = h_b + 2 \) and \( h_b = h_c + 2 \):
\[ h_a = (h_c + 2) + 2 = h_c + 4 \quad \Rightarrow \quad |h_a - h_c| = 4 \quad \text{(not equal to 2)} \]
2. If \( h_a = h_b + 2 \) and \( h_b = h_c - 2 \):
\[ h_a = (h_c - 2) + 2 = h_c \quad \Rightarrow \quad |h_a - h_c| = 0 \quad \text{(not equal to 2)} \]
3. If \( h_a = h_b - 2 \) and \( h_b = h_c + 2 \):
\[ h_a = (h_c + 2) - 2 = h_c \quad \Rightarrow \quad |h_a - h_c| = 0 \quad \text{(not equal to 2)} \]
4. If \( h_a = h_b - 2 \) and \( h_b = h_c - 2 \):
\[ h_a = (h_c - 2) - 2 = h_c - 4 \quad \Rightarrow \quad |h_a - h_c| = 4 \quad \text{(not equal to 2)} \]

In all cases, we find that \( |h_a - h_c| \) does not equal 2. Therefore, the relation \( R \) is **not transitive**.

### Conclusion
The relation \( R \) is:
- Not reflexive
- Symmetric
- Not transitive

Thus, \( R \) is symmetric but not an equivalence relation.