Let A = {ONGC, BHEL, SAIL, GAIL, IOCL} and R be a relation defined as ''two elements of A are related if they share exactly one letter''. The relation R, is

Options

A. anti-symmetric
B. only transitive
C. only symmetric
D. equivalence

Answer

To solve the problem, we need to analyze the relation \( R \) defined on the set \( A = \{ \text{ONGC, BHEL, SAIL, GAIL, IOCL} \} \) where two elements are related if they share exactly one letter. We will determine the properties of this relation: whether it is antisymmetric, symmetric, transitive, or an equivalence relation.

### Step 1: Identify the pairs in relation \( R \)

We need to find all pairs of elements in \( A \) that share exactly one letter.

- **BHEL and SAIL** share 'A'.
- **BHEL and GAIL** share 'B'.
- **BHEL and IOCL** share 'L'.
- **SAIL and GAIL** share 'A'.
- **GAIL and ONGC** share 'G'.
- **IOCL and BHEL** share 'L'.
- **SAIL and BHEL** share 'A'.
- **GAIL and BHEL** share 'B'.

From this analysis, we can list the pairs in relation \( R \):
- \( R = \{ (\text{BHEL, SAIL}), (\text{BHEL, GAIL}), (\text{BHEL, IOCL}), (\text{SAIL, GAIL}), (\text{GAIL, ONGC}), (\text{IOCL, BHEL}), (\text{SAIL, BHEL}), (\text{GAIL, BHEL}) \} \)

### Step 2: Check for Antisymmetry

A relation \( R \) is antisymmetric if for all \( a, b \in A \), if \( (a, b) \in R \) and \( (b, a) \in R \), then \( a = b \).

- In our relation \( R \), we can see that if \( (a, b) \in R \), it is not necessarily true that \( (b, a) \in R \) unless \( a = b \). For example, \( (\text{BHEL, SAIL}) \in R \) but \( (\text{SAIL, BHEL}) \) is not in \( R \) as they share only one letter. Thus, \( R \) is **not antisymmetric**.

### Step 3: Check for Symmetry

A relation \( R \) is symmetric if for all \( a, b \in A \), if \( (a, b) \in R \), then \( (b, a) \in R \).

- From our pairs, we see that for every pair \( (a, b) \) in \( R \), the reverse \( (b, a) \) is also present. For example, if \( (BHEL, SAIL) \in R \), then \( (SAIL, BHEL) \) is also in \( R \). Thus, \( R \) is **symmetric**.

### Step 4: Check for Transitivity

A relation \( R \) is transitive if for all \( a, b, c \in A \), if \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \).

- We need to check if there are any instances where \( (a, b) \in R \) and \( (b, c) \in R \) lead to \( (a, c) \in R \). For example, if \( (BHEL, SAIL) \in R \) and \( (SAIL, GAIL) \in R \), we do not find \( (BHEL, GAIL) \) in \( R \). Thus, \( R \) is **not transitive**.

### Conclusion

Based on the analysis:
- The relation \( R \) is **not antisymmetric**.
- The relation \( R \) is **symmetric**.
- The relation \( R \) is **not transitive**.

Thus, the only property that holds for relation \( R \) is that it is **symmetric**.

### Final Answer
The relation \( R \) is **only symmetric**.

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