Objective: Construct relations R on set A = {1, 2, 3} satisfying four criteria and count unique valid relations.
The criteria are:
1. R must contain (1, 2) and (1, 3).
2. R must be reflexive.
3. R must be symmetric.
4. R must NOT be transitive.
Analysis:
Let R be a relation on A.
For reflexivity (Condition 2), R must contain {(1, 1), (2, 2), (3, 3)}.
For symmetry (Condition 3) and Condition 1:
Since (1, 2) $\in$ R, then (2, 1) $\in$ R.
Since (1, 3) $\in$ R, then (3, 1) $\in$ R.
The minimal relation R_min satisfying Conditions 1, 2, and 3 is:
\[ R_{min} = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1)\} \]
Now, check for transitivity (Condition 4):
A relation is transitive if (a, b) $\in$ R and (b, c) $\in$ R implies (a, c) $\in$ R.
Consider (2, 1) $\in$ R_min and (1, 3) $\in$ R_min. For transitivity, (2, 3) must be in R_min. (2, 3) $otin$ R_min.
Consider (3, 1) $\in$ R_min and (1, 2) $\in$ R_min. For transitivity, (3, 2) must be in R_min. (3, 2) $otin$ R_min.
Thus, R_min is NOT transitive, satisfying Condition 4.
Are there other possibilities?
The pairs not in R_min are (2, 3) and (3, 2). To maintain symmetry, if we add (2, 3), we must also add (3, 2).
Let R' = R_min $\cup$ {(2, 3), (3, 2)}.
R' = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1), (2, 3), (3, 2)}. This is the universal relation A $\times$ A.
Check transitivity for R':
(2, 1) $\in$ R' and (1, 3) $\in$ R'. This implies (2, 3) $\in$ R'. R' contains (2, 3).
The universal relation is always transitive. Therefore, adding {(2, 3), (3, 2)} makes the relation transitive, violating Condition 4.
Conclusion:
Only R_min satisfies all four conditions.
Therefore, there is exactly 1 such relation.
Result: 1