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 \( \boxed{7} \).

Answer 2

Step 1: Basic facts about relations. 
Set \( A = \{1, 2, 3\} \). A relation \( R \subseteq A \times A \) is:

• Reflexive ⟹ all pairs \( (1,1), (2,2), (3,3) \) must be in \( R \).
• Transitive ⟹ whenever \( (a,b) \) and \( (b,c) \) are in \( R \), then \( (a,c) \) must also be in \( R \).
• Not symmetric ⟹ there exists at least one pair \( (x,y) \in R \) such that \( (y,x) \notin R \).

 

Step 2: Start with the given pairs.
The relation must contain \( (1,2) \) and \( (2,3) \). Because the relation must be transitive: \[ (1,2) \text{ and } (2,3) \implies (1,3) \] must also be included.

Step 3: Include reflexive pairs.
Reflexivity requires: \[ (1,1), (2,2), (3,3) \] must be in the relation.

So, the minimal reflexive and transitive relation so far is: \[ R_0 = \{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)\}. \]

Step 4: Check symmetry condition.
If the relation were symmetric, it must include the reverse of all non-diagonal pairs: \[ (2,1), (3,2), (3,1) \] But we require not symmetric, meaning at least one of these should be missing.

Step 5: Add possible reverse pairs ensuring transitivity.
Let’s test all combinations of adding some of these three possible reverse pairs, while preserving transitivity.

Start with \( R_0 \). The possible additional pairs are: \( (2,1), (3,2), (3,1) \).

Transitivity check conditions:

• If \( (2,1) \) and \( (1,2) \) are present → must have \( (2,2) \), already present.
• If \( (3,2) \) and \( (2,3) \) are present → must have \( (3,3) \), already present.
• If \( (3,1) \) and \( (1,3) \) are present → must have \( (3,3) \), already present.
• Also, if \( (2,1) \) and \( (1,3) \) are present → must have \( (2,3) \), already there.
• If \( (3,2) \) and \( (2,1) \) are present → must have \( (3,1) \).

So, when we include both \( (2,1) \) and \( (3,2) \), transitivity forces us to also include \( (3,1) \).

 

Step 6: List all transitive extensions of \( R_0 \).
Possible combinations of added pairs (from \( (2,1),(3,2),(3,1) \)) that are transitive:

1. No additional pairs → \( R_0 \) itself.
2. \( + (2,1) \)
3. \( + (3,2) \)
4. \( + (3,1) \)
5. \( + (2,1), (3,2), (3,1) \) (since both 2→1 and 3→2 force 3→1)
6. \( + (2,1), (3,1) \)
7. \( + (3,2), (3,1) \)

These 7 satisfy transitivity and reflexivity.

 

Step 7: Check which are not symmetric.
All 7 contain at least one directed pair without its reverse, so all are not symmetric.

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Final Answer:

\[ \boxed{7} \]