Question

Let \( R \) be a relation on \( \{1,2,3\} \) defined by \[ R=\{(1,1),(2,2),(3,3),(1,2)\} \] Identify the properties satisfied by \(R\).

• A. Symmetric only
• B. Reflexive and Transitive
• C. Equivalence relation
• D. None of these

Hint:

If a relation contains all identity pairs and very few cross-pairs, it is often reflexive and transitive but fails symmetry.

Answer 1

The Correct Option is B

Concept: A relation is:
• Reflexive if every element is related to itself.
• Symmetric if \((a,b)\in R \Rightarrow (b,a)\in R\).
• Transitive if \((a,b)\in R\) and \((b,c)\in R\Rightarrow(a,c)\in R\).

Step 1: Check reflexivity:
Since \[ (1,1),(2,2),(3,3)\in R \] the relation is reflexive.

Step 2: Check symmetry:
We have: \[ (1,2)\in R \] but \[ (2,1)\notin R \] Hence the relation is not symmetric.

Step 3: Check transitivity:
The only nontrivial ordered pair is \((1,2)\). Since there is no pair beginning with \(2\) except \((2,2)\), transitivity is satisfied: \[ (1,2),(2,2)\Rightarrow(1,2)\in R \] Thus the relation is transitive.