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. Consequently, for A = {1, 2, 3}, the pairs \( (1,1), (2,2), \) and \( (3,3) \) must be included.
Given the inclusion of \( (1,2) \) and \( (2,3) \), transitivity mandates the inclusion of \( (1,3) \).
Step 2: Determine the count of valid relations.
The elements \( (2,1) \) and \( (3,2) \) are potential additions but must be excluded to avoid symmetry.
The number of relations that meet the criteria of reflexivity and transitivity, while excluding symmetry, is calculated as:
