The number of relations on the set {1,2,3}, containing at most 6 elements including (1,2), which are reflexive and transitive but not symmetric, is \_\_\_\_.
Answer: 6
Solutions · 1
Official solution
Reflexivity + transitivity + containing (1,2) forces the base relation
\[R_0 = \{(1,1),(2,2),(3,3),(1,2)\}\]
(4 elements). At most 2 more pairs (to stay within 6 elements) can be chosen from {(2,1),(1,3),(3,1),(2,3),(3,2)}, while keeping the relation transitive and not symmetric. Careful enumeration (checking every transitive chain each addition creates) gives exactly the following 6 valid relations:
1. R0 (no additions)
2. R0∪{(1,3)}
3. R0∪{(3,2)}
4. R0∪{(1,3),(3,2)}
5. R0∪{(3,1),(3,2)}
6. R0∪{(1,3),(2,3)}
Each remains transitive (all forced consequence-pairs are present) while at least one pair still lacks its symmetric reverse. Any other combination either breaks transitivity or forces symmetry. Hence the count is 6.