Question

Determine which one of the following relations on \( X = \{1, 2, 3, 4\} \) is not transitive.

• A. \( R = R^2 \), the empty relation
• B. \( R = X \times X \), the universal relation
• C. \( R = \{(1, 1), (2, 2)\} \)
• D. \( R = \{(1, 2), (2, 3), (3, 4)\} \)

Hint:

To check transitivity, verify if for every pair \( (a, b) \) and \( (b, c) \), the pair \( (a, c) \) is also in the relation.

Answer 1

The Correct Option is C

Step 1: Understand transitivity. A relation \( R \) is transitive if, for any \( a, b, c \), whenever \( (a, b) \in R \) and \( (b, c) \in R \), we also have \( (a, c) \in R \).
Step 2: Conclusion. The relation \( R = \{(1, 1), (2, 2)\} \) is not transitive.