Question

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 ______.

• A. 10
• B. 12
• C. 8
• D. 15

Answer 1

The Correct Option is A

Given set - \(\{1, 2, 3, 4\}\).

To form the smallest equivalence relation on this set that includes \((1, 2)\) and \((1, 3)\), we need to ensure that \(R\) is reflexive, symmetric, and transitive.  

Step 1. Reflexive pairs: \((1, 1), (2, 2), (3, 3), (4, 4)\)
Step 2. Pairs to satisfy given conditions and transitivity:
  - Since \((1, 2) \in R\) and \((1, 3) \in R\), we need \((2, 3) \in R\) for transitivity.  
  - For symmetry, include \((2, 1), (3, 1), (3, 2)\).  

Step 3. Final set of pairs: \(R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (1, 3), (3, 1), (2, 3), (3, 2)\}\).

Thus, the number of elements in \(R\) is 10.  

The Correct Answer is: 10