Question

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 \_\_\_\_\_\_.

Hint:

When dealing with reflexive and transitive relations, enforce required pairs first, then check minimal conditions for additional elements.

Answer 1

Correct Answer: 7

Step 1: Define the reflexive and transitive conditions. A relation is reflexive if it contains \( (x,x) \) for all \( x \in A \), meaning it must have \( (1,1), (2,2), (3,3) \). Since \( (1,2) \) and \( (2,3) \) are included, transitivity requires \( (1,3) \) to be included. Step 2: Count valid relations. The possible additional elements are \( (2,1) \) and \( (3,2) \), which must be avoided to prevent symmetry. The valid relations satisfying reflexivity and transitivity but not symmetry are counted, giving: \[ 7. \] Thus, the answer is \( \boxed{7} \).