3. Let 𝑅 be defined on ℝ by (𝑥, 𝑦) ∈ 𝑅 if |𝑥 − 𝑦| ≥ 1. Check if 𝑅 is (i) reflexive, (ii) symmetric, (iii) antisymmetric, (iv) transitive.

To determine the properties of the relation \( R \) defined on \( \mathbb{R} \) by \( (x, y) \in R \) if \( |x - y| \geq 1 \), we will check each property:

### (i) Reflexive
A relation is reflexive if for all \( x \in \mathbb{R} \), \( (x, x) \in R \).  
- Here, \( |x - x| = 0 \), which is not greater than or equal to 1.  
- Therefore, \( R \) is **not reflexive**.

### (ii) Symmetric
A relation is symmetric if whenever \( (x, y) \in R \), then \( (y, x) \in R \).  
- If \( |x - y| \geq 1 \), then \( |y - x| = |x - y| \geq 1 \) as well.  
- Therefore, \( R \) is **symmetric**.

### (iii) Antisymmetric
A relation is antisymmetric if whenever \( (x, y) \in R \) and \( (y, x) \in R \), then \( x = y \).  
- Since \( |x - y| \geq 1 \) and \( |y - x| \geq 1 \) both imply \( x \neq y \), \( R \) cannot be antisymmetric.  
- Therefore, \( R \) is **not antisymmetric**.

### (iv) Transitive
A relation is transitive if whenever \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \in R \).  
- Consider \( x = 0 \), \( y = 2 \), and \( z = 3 \):
  - \( |0 - 2| = 2 \geq 1 \) (so \( (0, 2) \in R \))
  - \( |2 - 3| = 1 \geq 1 \) (so \( (2, 3) \in R \))
  - But \( |0 - 3| = 3 \geq 1 \) (so \( (0, 3) \in R \)). 

- However, consider \( x = 0 \), \( y = 2 \), \( z = 1 \):
  - \( |0 - 2| = 2 \geq 1 \) (so \( (0, 2) \in R \))
  - \( |2 - 1| = 1 \geq 1 \) (so \( (2, 1) \in R \))
  - But \( |0 - 1| = 1 \), so \( (0, 1) \notin R \).

- Since we found a counterexample, \( R \) is **not transitive**.

### Summary
- **Reflexive**: No
- **Symmetric**: Yes
- **Antisymmetric**: No
- **Transitive**: No

Would you like more details on any specific property, or do you have any questions? Here are some related questions you might consider:

1. What are examples of reflexive relations on \( \mathbb{R} \)?
2. Can you provide an example of a relation that is both symmetric and transitive?
3. How do we determine if a relation is an equivalence relation?
4. What other properties can be used to classify relations?
5. How does the definition of \( R \) change if the condition is \( |x - y| > 1 \)? 

**Tip:** When analyzing relations, always consider counterexamples to disprove properties like transitivity or antisymmetry.

Let R be defined on ℝ by (x, y) ∈ R if |x − y| ≥ 1. Check if R is (i) reflexive, (ii) symmetric, (iii) antisymmetric, (iv) transitive.

Explore the properties of the relation R defined on ℝ by |x - y| ≥ 1. This analysis includes checks for reflexivity, symmetry, antisymmetry, and transitivity, providing clear insights for students in Grades 10-12.

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