Final answer:
There is only one relation on the set {1, 2, 3} that contains (1, 2) and (2, 3) and is reflexive and transitive but not symmetric. This relation includes the pairs (1, 1), (2, 2), (3, 3), (1, 2), (2, 3), and (1, 3).
Step-by-step explanation:
The question is asking for the number of reflexive and transitive relations, on the set {1, 2, 3}, which also include the pairs (1, 2) and (2, 3) but are not symmetric. For a relation to be reflexive, every element must relate to itself; thus, the pairs (1, 1), (2, 2), and (3, 3) must be part of the relation. As it must be transitive, including (1, 2) and (2, 3) in the relation implies that (1, 3) must also be included because if 1 is related to 2, and 2 is related to 3, then 1 must be related to 3 for the relation to be transitive.
To ensure the relation is not symmetric, we must not include (2, 1) or (3, 2) because the given pairs (1, 2) and (2, 3) would then require these for symmetry. Therefore, the reflexive and transitive relations on the set that include (1, 2) and (2, 3) but are not symmetric are those that contain (1, 1), (2, 2), (3, 3), (1, 2), (1, 3), and (2, 3) as mandatory elements. Any other pairs including (2, 1) or (3, 2) would make the relation symmetric, which is not allowed, and any other pairs involving (3, 1) are unnecessary for the relation to be reflexive or transitive and can be freely included or excluded.
Therefore, we can have relations where the non-mandatory pairs (3, 1) may or may not be present. There is only one possible combination where the relation is reflexive and transitive but not symmetric. Hence there is only one relation that satisfies the conditions.