If \( R \) is the smallest equivalence relation on the set \( \{1, 2, 3, 4\} \) such that \( \{(1,2), (1,3)\} \subseteq R \), then the number of elements in \( R \) is ______.
To determine the minimal equivalence relation \( R \) on the set \( \{1, 2, 3, 4\} \) that includes the pairs \( \{(1,2), (1,3)\} \), \( R \) must be reflexive, symmetric, and transitive.
Step 1: Reflexivity
For \( R \) to be reflexive, it must contain \( \{(1,1), (2,2), (3,3), (4,4)\} \).
Step 2: Incorporating Given Pairs and Symmetry
The given pairs are \( (1,2) \) and \( (1,3) \).
Symmetry mandates that if \( (a,b) \in R \), then \( (b,a) \in R \). Thus, \( (2,1) \) and \( (3,1) \) are added.
Step 3: Ensuring Transitivity
Transitivity requires that if \( (a,b) \in R \) and \( (b,c) \in R \), then \( (a,c) \in R \).
With \( (1,2) \) and \( (2,1) \), we confirm \( (1,1) \) (already present).
With \( (1,3) \) and \( (3,1) \), we confirm \( (1,1) \).
Consider the chain \( (1,2) \) and \( (2,1) \), no new pairs added.
Consider the chain \( (1,3) \) and \( (3,1) \), no new pairs added.
Consider the chain \( (2,1) \) and \( (1,3) \), implying \( (2,3) \) and its symmetric counterpart \( (3,2) \) must be added.
Full closure ensures all transitive implications are covered.
Final Equivalence Relation \( R \)
The complete set \( R \) comprises these 10 pairs:
Reflexive: (1,1), (2,2), (3,3), (4,4)
Symmetric from given: (1,2), (2,1), (1,3), (3,1)
Transitive additions: (2,3), (3,2)
Therefore, the equivalence relation \( R \) has 10 elements.
