Let \( A = \{1,2,3\} \). The number of relations on \( A \), containing \( (1,2) \) and \( (2,3) \), which are reflexive and transitive but not symmetric, is ________.
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When dealing with reflexive and transitive relations, enforce required pairs first, then check minimal conditions for additional elements.
Step 1: Establish the conditions for reflexivity and transitivity. A relation is reflexive if, for every element \( x \) in set \( A \), the pair \( (x,x) \) is present. For \( A = \{1, 2, 3\} \), this necessitates the inclusion of \( (1,1), (2,2), (3,3) \). Given that \( (1,2) \) and \( (2,3) \) are present, transitivity mandates the inclusion of \( (1,3) \).
Step 2: Determine the count of valid relations. To maintain a lack of symmetry, the pairs \( (2,1) \) and \( (3,2) \) must be excluded.
The relations that fulfill reflexivity and transitivity but not symmetry are enumerated. The count is as follows: \[ 7. \] Therefore, the final answer is \( \boxed{7} \).
