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.