Let A = 1, 2, 3, 4 . The number of relations on A which are reflexive and symmetric but not transitive, and which contain the ordered pair (1, 2) , is ______.

Step-by-step solution

Let A = 1, 2, 3, 4 . We need to find the number of relations on A that are reflexive, symmetric, and contain (1, 2) , but are not transitive. First, let us count the total number of relations on A that are reflexive, symmetric, and contain (1, 2) . For a relation to be reflexive, it must contain all diagonal pairs: (1, 1), (2, 2), (3, 3), (4, 4) . For it to be symmetric, if it contains (a, b) , it must contain (b, a) . Since (1, 2) is in the relation, (2, 1) must also be in it. The remaining non-diagonal pairs can