Question

Let \(R=\{(3,3),(6,6),(9,9),(12,12),(6,12),(3,9),(3,12),(3,6)\}\) be a relation on the set \(A=\{3,6,9,12\}\). The relation is

• A. An equivalence relation
• B. Reflexive and symmetric only
• C. Reflexive and transitive only
• D. Reflexive only

Hint:

For relations on a set:
Reflexive: \((a,a)\) for all elements
Symmetric: \((a,b)\Rightarrow(b,a)\)
Transitive: \((a,b)\) and \((b,c)\Rightarrow(a,c)\) An equivalence relation must satisfy \textbf{all three}.

Answer 1

The Correct Option is D

Step 1: Check reflexive property A relation on set \(A\) is reflexive if \((a,a)\in R\) for all \(a\in A\). Given: \[ (3,3),(6,6),(9,9),(12,12)\in R \] Hence, \(R\) is reflexive.
Step 2: Check symmetric property A relation is symmetric if \((a,b)\in R \Rightarrow (b,a)\in R\). Here, \[ (3,6)\in R \quad \text{but} \quad (6,3)\notin R \] Hence, \(R\) is not symmetric.
Step 3: Check transitive property A relation is transitive if \((a,b)\in R\) and \((b,c)\in R \Rightarrow (a,c)\in R\). Consider: \[ (9,3)\in R \ \text{and}\ (3,6)\in R \] But, \[ (9,6)\notin R \] Hence, \(R\) is not transitive.
Step 4: Since the relation is reflexive but neither symmetric nor transitive, it is reflexive only.