To determine whether the relations \( R_1 \) and \( R_2 \) are equivalence relations, we need to check the three properties: reflexivity, symmetry, and transitivity.
### Step 1: Analyze \( R_1 \)
**Definition of \( R_1 \)**:
\[ R_1 = \{(a, b) \in \mathbb{N} \times \mathbb{N} : |a - b| \leq 13\} \]
**Check Reflexivity**:
- For reflexivity, we need to check if \( (a, a) \in R_1 \) for all \( a \in \mathbb{N} \).
- \( |a - a| = 0 \leq 13 \) (True)
- Thus, \( R_1 \) is reflexive.
**Check Symmetry**:
- For symmetry, we need to check if \( (a, b) \in R_1 \) implies \( (b, a) \in R_1 \).
- If \( |a - b| \leq 13 \), then \( |b - a| = |a - b| \leq 13 \) (True)
- Thus, \( R_1 \) is symmetric.
**Check Transitivity**:
- For transitivity, we need to check if \( (a, b) \in R_1 \) and \( (b, c) \in R_1 \) implies \( (a, c) \in R_1 \).
- Assume \( |a - b| \leq 13 \) and \( |b - c| \leq 13 \).
- Using the triangle inequality, \( |a - c| \leq |a - b| + |b - c| \leq 13 + 13 = 26 \).
- However, this does not guarantee \( |a - c| \leq 13 \).
- Thus, \( R_1 \) is not transitive.
### Conclusion for \( R_1 \):
Since \( R_1 \) is reflexive and symmetric but not transitive, it is **not an equivalence relation**.
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### Step 2: Analyze \( R_2 \)
**Definition of \( R_2 \)**:
\[ R_2 = \{(a, b) \in \mathbb{N} \times \mathbb{N} : |a - b| \neq 13\} \]
**Check Reflexivity**:
- For reflexivity, we check if \( (a, a) \in R_2 \) for all \( a \in \mathbb{N} \).
- \( |a - a| = 0 \neq 13 \) (True)
- Thus, \( R_2 \) is reflexive.
**Check Symmetry**:
- For symmetry, we check if \( (a, b) \in R_2 \) implies \( (b, a) \in R_2 \).
- If \( |a - b| \neq 13 \), then \( |b - a| = |a - b| \neq 13 \) (True)
- Thus, \( R_2 \) is symmetric.
**Check Transitivity**:
- For transitivity, we need to check if \( (a, b) \in R_2 \) and \( (b, c) \in R_2 \) implies \( (a, c) \in R_2 \).
- Assume \( |a - b| \neq 13 \) and \( |b - c| \neq 13 \).
- It is possible that \( |a - c| = 13 \) (for example, if \( a = 1, b = 2, c = 15 \)).
- Thus, \( R_2 \) is not transitive.
### Conclusion for \( R_2 \):
Since \( R_2 \) is reflexive and symmetric but not transitive, it is **not an equivalence relation**.
### Final Conclusion:
Both \( R_1 \) and \( R_2 \) are **not equivalence relations**.
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