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 ______.
- 10
- 12
- 8
- 15
The Correct Option is A
Approach Solution - 1
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.
Approach Solution -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
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