Given:
- Set A: {a, b, c}
- Relation R: {(a, c)}
Assertion:
R = {(a, c)} is transitive.
Reason:
A singleton relation is transitive.
Analysis:
Transitivity Definition:
A relation \( R \) on a set \( A \) is transitive if for all \( x, y, z \in A \), whenever \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \in R \).
Given Relation R:
- \( R \) contains only one pair: {(a, c)}.
Checking Transitivity:
- There are no pairs \( (x, y) \) and \( (y, z) \) to check other than the given pair \((a, c)\).
- Since there are no counterexamples, \( R \) is trivially transitive.
Conclusion:
- Assertion: True. \( R \) is transitive.
- Reason: True. A singleton relation is transitive because it has no other pairs to consider.
Summary:
- Assertion: The relation \( R = \{(a, c)\} \) is transitive.
- Reason: A singleton relation is transitive.
Both the assertion and the reason are true, and the reason correctly supports the assertion.
