Let X be the set of all engineering colleges in a state of Indian Republic and R be a relation on X defined as two colleges are related iff they are affiliated to the same university, then R is

Options

A. only reflexive
B. only symmetric
C. only transitive
D. equivalence

Answer

To determine the properties of the relation \( R \) defined on the set \( X \) of all engineering colleges in a state of Indian Republic, where two colleges are related if they are affiliated to the same university, we will check if the relation is reflexive, symmetric, and transitive. If it satisfies all three properties, it is an equivalence relation.

### Step-by-Step Solution:

1. **Check for Reflexivity**:
- A relation \( R \) is reflexive if for every element \( a \in X \), the pair \( (a, a) \) is in \( R \).
- In this case, if college \( a \) is affiliated to a university, say \( U \), then it is obviously affiliated to \( U \) itself.
- Therefore, \( (a, a) \) is in \( R \) for all colleges \( a \).
- **Conclusion**: \( R \) is reflexive.

2. **Check for Symmetry**:
- A relation \( R \) is symmetric if whenever \( (a, b) \in R \), then \( (b, a) \in R \).
- If college \( a \) is affiliated to university \( U \) and college \( b \) is also affiliated to the same university \( U \), then \( (a, b) \) is in \( R \).
- Since both colleges are affiliated to the same university, it follows that \( (b, a) \) is also in \( R \).
- **Conclusion**: \( R \) is symmetric.

3. **Check for Transitivity**:
- A relation \( R \) is transitive if whenever \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \).
- If college \( a \) is affiliated to university \( U_1 \) and college \( b \) is affiliated to \( U_1 \), and college \( b \) is also affiliated to university \( U_2 \) and college \( c \) is affiliated to \( U_2 \), then both \( a \) and \( c \) must be affiliated to the same university \( U_1 \) or \( U_2 \).
- Therefore, if \( a \) is affiliated to the same university as \( b \) and \( b \) is affiliated to the same university as \( c \), then \( a \) must also be affiliated to the same university as \( c \).
- **Conclusion**: \( R \) is transitive.

Since \( R \) is reflexive, symmetric, and transitive, we conclude that \( R \) is an equivalence relation.

### Final Conclusion:
The relation \( R \) is an equivalence relation.

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