. A relation R is defined on Z by a R b if |a − b| ∈ {0} ∪ {2 n : n ∈ Z, n ≥ 0}. (a) Give an example of two integers that are related by R and two integers that are not. (b) Which of the properties reflexive, symmetric, transitive does R possess?

We are given a relation \( R \) defined on the set of integers \( \mathbb{Z} \). The relation is defined as:

\[
a \, R \, b \quad \text{if} \quad |a - b| \in \{0\} \cup \{2n : n \in \mathbb{Z}, n \geq 0\}
\]

This means that two integers \( a \) and \( b \) are related by \( R \) if the absolute difference \( |a - b| \) is either \( 0 \) or a non-negative even integer.

### (a) Examples

1. **Two integers that are related by \( R \):**
   - Let \( a = 4 \) and \( b = 2 \). Then:
     \[
     |4 - 2| = 2 \quad (\text{which is an even integer}).
     \]
     Since \( 2 \in \{2n : n \in \mathbb{Z}, n \geq 0\} \), the integers 4 and 2 are related by \( R \).

2. **Two integers that are not related by \( R \):**
   - Let \( a = 3 \) and \( b = 1 \). Then:
     \[
     |3 - 1| = 2 \quad (\text{which is an even integer, so they are related}).
     \]
     Instead, let's pick \( a = 3 \) and \( b = 0 \). Then:
     \[
     |3 - 0| = 3 \quad (\text{which is not even, so they are not related by } R).
     \]

### (b) Reflexive, Symmetric, and Transitive Properties

1. **Reflexive:**
   - A relation \( R \) is reflexive if for all \( a \in \mathbb{Z} \), \( a \, R \, a \).
   - Here, \( |a - a| = 0 \), and since \( 0 \in \{0\} \cup \{2n : n \in \mathbb{Z}, n \geq 0\} \), the relation \( R \) is **reflexive**.

2. **Symmetric:**
   - A relation \( R \) is symmetric if \( a \, R \, b \) implies \( b \, R \, a \).
   - Since \( |a - b| = |b - a| \), the absolute difference is the same in both directions. Therefore, \( R \) is **symmetric**.

3. **Transitive:**
   - A relation \( R \) is transitive if \( a \, R \, b \) and \( b \, R \, c \) imply \( a \, R \, c \).
   - Suppose \( |a - b| \in \{0\} \cup \{2n : n \in \mathbb{Z}, n \geq 0\} \) and \( |b - c| \in \{0\} \cup \{2n : n \in \mathbb{Z}, n \geq 0\} \). However, this does not guarantee that \( |a - c| \in \{0\} \cup \{2n : n \in \mathbb{Z}, n \geq 0\} \). For example:
     - Let \( a = 0 \), \( b = 2 \), and \( c = 3 \).
     - \( |0 - 2| = 2 \) and \( |2 - 3| = 1 \), but \( |0 - 3| = 3 \), which is not even.
   - Thus, \( R \) is **not transitive**.

### Summary of Properties:
- Reflexive: **Yes**
- Symmetric: **Yes**
- Transitive: **No**

Would you like further details or clarifications on any part?

Here are 5 related questions to expand your understanding:
1. How would the relation \( R \) change if we modified the condition to \( |a - b| \in \{2n : n \in \mathbb{Z}\} \)?
2. What does it mean for a relation to be an equivalence relation, and does \( R \) qualify as one?
3. Can you provide another example of a symmetric but non-transitive relation?
4. How does the concept of symmetry in relations differ from symmetry in geometry?
5. What are the real-life applications of symmetric and reflexive relations?

**Tip:** To check if a relation is transitive, test it with three elements where the relation holds pairwise, and check if it holds for the entire set.

A relation R is defined on Z by a R b if |a − b| ∈ {0} ∪ {2n : n ∈ Z, n ≥ 0}. (a) Give an example of two integers that are related by R and two integers that are not. (b) Which of the properties reflexive, symmetric, transitive does R possess?

Explore the relation R on the set of integers defined by |a − b| ∈ {0} ∪ {2n : n ∈ Z, n ≥ 0}. This problem delves into examples of related and non-related integers and examines whether the relation is reflexive, symmetric, and transitive. Suitable for high school students (Grades 9-12).

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