Question

The number of relations on the set $ A = \{1, 2, 3\} $ containing at most 6 elements including $ (1, 2) $, which are reflexive and transitive but not symmetric, is:

Hint:

When dealing with reflexive and transitive relations, be sure to include the required reflexive pairs and check for transitivity by including necessary pairs. Avoid including pairs that would make the relation symmetric if the condition specifies it should not be symmetric.

Answer 1

Correct Answer: 6

Given the set \( A = \{1, 2, 3\} \) and the relation \( R \) containing \( (1, 1), (2, 2), (3, 3), (1, 2) \). The remaining possible pairs for a relation on \( A \) are \( (2, 1), (2, 3), (1, 3), (3, 1), (3, 2) \). We aim to find the number of relations on \( A \) that have at most 6 elements, are reflexive and transitive, but not symmetric.

Step 1: Relations with exactly 4 elements

A relation with 4 elements must include the reflexive pairs \( (1, 1), (2, 2), (3, 3) \). We must select one additional element from the set \( \{ (2, 1), (2, 3), (1, 3), (3, 1), (3, 2) \} \). To maintain transitivity, choosing \( (1, 2) \) is the only valid option, resulting in the relation \( \{(1, 1), (2, 2), (3, 3), (1, 2)\} \). This relation is reflexive and transitive but not symmetric.

Therefore, there is 1 way to form a relation with 4 elements.

Step 2: Relations with exactly 5 elements

These relations must include \( (1, 1), (2, 2), (3, 3) \). We need to choose two additional elements from \( \{ (2, 1), (2, 3), (1, 3), (3, 1), (3, 2) \} \). The valid combinations that maintain transitivity are \( \{(1, 3), (3, 2)\} \) or \( \{(3, 1), (2, 3)\} \). This yields two distinct relations:

1. \( \{(1, 1), (2, 2), (3, 3), (1, 3), (3, 2)\} \)
2. \( \{(1, 1), (2, 2), (3, 3), (3, 1), (2, 3)\} \)

Both are reflexive and transitive but not symmetric.

Thus, there are 2 ways to form a relation with 5 elements.

Step 3: Relations with exactly 6 elements

These relations include all reflexive pairs \( (1, 1), (2, 2), (3, 3) \) and three additional pairs from \( \{ (2, 1), (2, 3), (1, 3), (3, 1), (3, 2) \} \). To ensure the relation is transitive and not symmetric, we can form the following three relations:

1. \( \{(1, 1), (2, 2), (3, 3), (2, 3), (1, 3), (3, 2)\} \)
2. \( \{(1, 1), (2, 2), (3, 3), (2, 3), (3, 1), (1, 3)\} \)
3. \( \{(1, 1), (2, 2), (3, 3), (3, 2), (1, 3), (3, 1)\} \)

Note: The example provided in the original text for 6 elements was not exhaustive and contained duplicates and invalid combinations according to the stated conditions. The listed relations above are valid and satisfy the criteria.

Thus, there are 3 ways to form a relation with 6 elements.

Final Answer: The total number of ways is the sum of ways for each case: \( 1 + 2 + 3 = 6 \).