Question

Prove that the relation given by \(R=\{(1,1),(2,2),(3,3),(1,2),(2,3)\}\) in the set \(\{1,2,3\}\) is reflexive but neither symmetric nor transitive.

Hint:

Check the three properties directly by listing the pairs.

Answer 1

Step 1: Reflexive test via all diagonal pairs:
All three diagonal pairs \((1,1),(2,2),(3,3)\) are present, so \(R\) is reflexive by definition.

Step 2: Producing one counterexample each:
For symmetry: \((1,2)\in R\) but \((2,1)\notin R\) is enough to disprove symmetry.

Step 3: For transitivity:
\((1,2)\in R,(2,3)\in R\) would require \((1,3)\in R\) for transitivity, but \((1,3)\notin R\), disproving transitivity.

Final Answer:
\[ \boxed{\text{Reflexive only}} \]