Question

Consider the set \( M = \{1,2,3\} \) along with the relation \( R = \{(1,2), (1,1), (3,1), (3,4), (3,3), (4,3)\ \). Which of the following statements is true?

• A. The relation is symmetric but not transitive
• B. The relation is transitive but not symmetric
• C. The relation is both symmetric and transitive
• D. The relation is neither symmetric nor transitive
• E. The relation is reflexive

Hint:

Always check symmetry using one counterexample, and transitivity using chain pairs like \((a,b),(b,c)\).

Answer 1

The Correct Option is D

Concept: For a relation \(R\):
• Symmetric: If \( (a,b) \in R \Rightarrow (b,a) \in R \)
• Transitive: If \( (a,b), (b,c) \in R \Rightarrow (a,c) \in R \)
• Reflexive: If \( (a,a) \in R \) for all \( a \in M \)

Step 1: Check symmetry. Given: \[ (1,2) \in R \text{but} (2,1) \notin R \] Thus symmetry condition fails. \[ \Rightarrow R \text{ is not symmetric} \]

Step 2: Check transitivity. Observe: \[ (3,4) \in R \text{and} (4,3) \in R \] So for transitivity: \[ (3,3) \in R \text{(which exists)} \] Now check: \[ (1,2) \text{ exists but no } (2,x) \text{ pair} \] However: \[ (3,1) \in R \text{and} (1,2) \in R \] Then transitivity requires: \[ (3,2) \in R \] But \( (3,2) \notin R \) Thus transitivity fails. \[ \Rightarrow R \text{ is not transitive} \]

Step 3: Check reflexivity. Set \(M = \{1,2,3\}\) For reflexive: \[ (1,1), (2,2), (3,3) \in R \] But \( (2,2) \notin R \) \[ \Rightarrow R \text{ is not reflexive} \]