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

To find the smallest equivalence relation \( R \) on the set \( \{1, 2, 3, 4\} \) that includes the pairs \( \{(1,2), (1,3)\} \), we have to ensure that \( R \) satisfies the properties of an equivalence relation: reflexivity, symmetry, and transitivity.

Step 1: Reflexivity 

• For \( R \) to be reflexive, each element in the set must be related to itself. Therefore, \( R \) must include \( \{(1,1), (2,2), (3,3), (4,4)\} \).

Step 2: Including Given Pairs and Ensuring Symmetry

• We include \( (1,2) \) and \( (1,3) \) as given.
• Symmetry requires that if \( (a,b) \) is in \( R \), then \( (b,a) \) must also be in \( R \). Therefore, we add \( (2,1) \) and \( (3,1) \).

Step 3: Ensuring Transitivity

• Transitivity requires that if \( (a,b) \) and \( (b,c) \) are in \( R \), then \( (a,c) \) must be in \( R \).
• Since we have \( (1,2) \) and \( (2,1) \), we should have \( (1,1) \) (already included as part of reflexivity).
• Similarly, having \( (1,3) \) and \( (3,1) \) implies \( (1,1) \).
• Consider the chains: \( (1,2) \) and \( (2,3) \) (since \( (2,1) \) and \( (1,3) \)), add \( (2,3) \) and \( (3,2) \).
• Repeat with chains involving 3rd paths ensure all possible combinations: Add \( (2,3) \) and its mirror \( (3,2) \), which cover indirect transitive obligations. Still, verify: (Full closure).

Final Equivalence Relation \( R \)

• Collectively \( R \) will consist of these 10 pairs:
  • Reflexive pairs: (1,1), (2,2), (3,3), (4,4)
  • Symmetric additions: (1,2), (2,1), (1,3), (3,1)
  • Transitive necessary: (2,3), (3,2)

Thus, the equivalence relation \( R \) contains 10 elements.

Set of Pairs in \( R \)
(1,1), (2,2), (3,3), (4,4), (1,2), (2,1), (1,3), (3,1), (2,3), (3,2)

Therefore, the number of elements in \( R \) is 10, which matches the correct answer.

Answer 2

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