To determine the minimum number of elements that must be added to the relation \( R = \{(1, 2), (2, 3)\} \) on the set of natural numbers so that it becomes an equivalence relation, we need to ensure that the relation satisfies three properties: reflexivity, symmetry, and transitivity.
### Step-by-Step Solution:
1. **Understanding Equivalence Relation**:
An equivalence relation must be reflexive, symmetric, and transitive.
2. **Checking Reflexivity**:
For the relation to be reflexive, every element in the set must relate to itself. Here, we need to consider at least the elements 1, 2, and 3 (as they are present in the relation):
- We need to add (1, 1), (2, 2), and (3, 3) to make it reflexive.
- **Elements added for reflexivity**: 3 elements: \( (1, 1), (2, 2), (3, 3) \).
3. **Checking Symmetry**:
For the relation to be symmetric, if \( (a, b) \) is in the relation, then \( (b, a) \) must also be in the relation:
- From \( (1, 2) \), we need to add \( (2, 1) \).
- From \( (2, 3) \), we need to add \( (3, 2) \).
- **Elements added for symmetry**: 2 elements: \( (2, 1), (3, 2) \).
4. **Checking Transitivity**:
For the relation to be transitive, if \( (a, b) \) and \( (b, c) \) are in the relation, then \( (a, c) \) must also be in the relation:
- We have \( (1, 2) \) and \( (2, 3) \), which implies we need to add \( (1, 3) \).
- **Elements added for transitivity**: 1 element: \( (1, 3) \).
5. **Adding Symmetric Pairs for Transitivity**:
Since we added \( (1, 3) \), we also need to ensure symmetry for this pair:
- We need to add \( (3, 1) \).
- **Elements added for symmetry of transitive pair**: 1 element: \( (3, 1) \).
6. **Counting Total Elements Added**:
- From reflexivity: 3 elements
- From symmetry: 2 elements
- From transitivity: 1 element
- From symmetry of the transitive pair: 1 element
- **Total**: \( 3 + 2 + 1 + 1 = 7 \).
### Final Answer:
The minimum number of elements that must be added to the relation \( R \) to make it an equivalence relation is **7**.
