Question

The number of relations on \(A=\{1,2,3\}\) containing at most 6 elements including (1,2), that are reflexive and transitive but not symmetric is:

• A. \(4\)
• B. \(5\)
• C. \(6\)
• D. \(7\)

Hint:

In relation problems, always start with reflexive pairs and then apply transitivity closure step-by-step before counting.

Answer 1

The Correct Option is C

Concept: A reflexive relation on a 3-element set must contain: \[ (1,1),(2,2),(3,3) \] Transitivity restricts how extra ordered pairs can be added, since inclusion of certain pairs forces closure.

Step 1: Start with reflexive base set. \[ R_0=\{(1,1),(2,2),(3,3)\} \] So initial size = 3.

Step 2: Apply constraints. We must include (1,2), so: \[ R=\{(1,1),(2,2),(3,3),(1,2)\} \cup \text{possible extra pairs} \] Maximum size allowed = 6, so at most 2 more pairs can be added.

Step 3: Check transitivity restrictions. If (1,2) is included, then: - (2,1) cannot be included (would violate non-symmetry requirement) - Any addition like (2,3) or (1,3) must preserve closure Systematic checking of all valid closures gives only limited consistent structures.

Step 4: Count valid relations. After enumerating all transitive closures under given constraints, total valid relations = \[ \boxed{6} \]