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: Checking Reflexivity:
\(R\) is reflexive if \((a,a)\in R\) for every \(a\in\{1,2,3\}\). Here \((1,1),(2,2),(3,3)\) are all in \(R\), so \(R\) is reflexive.

Step 2: Checking Symmetry:
\(R\) is symmetric if \((a,b)\in R\Rightarrow(b,a)\in R\). But \((1,2)\in R\) while \((2,1)\notin R\), so \(R\) is not symmetric.

Step 3: Checking Transitivity:
\(R\) is transitive if \((a,b)\in R\) and \((b,c)\in R\Rightarrow(a,c)\in R\). Here \((1,2)\in R\) and \((2,3)\in R\), but \((1,3)\notin R\), so \(R\) is not transitive.

Final Answer:
\[ \boxed{R \text{ is reflexive, but not symmetric and not transitive}} \]