Given the relation $R = \{(x, y) : x - y \text{ is an even integer} \}$, we aim to determine the equivalence classes of $2$ (denoted as $\overline{2}$) and $-3$ (denoted as $\overline{-3}$).

Step 1: Understand the equivalence relation

• A relation $R$ is an equivalence relation if it is reflexive, symmetric, and transitive. The condition $x - y \text{ is an even integer}$ satisfies all three properties:
  1. Reflexive: For any $x$, $x - x = 0$ (even), so $(x, x) \in R$.
  2. Symmetric: If $x - y$ is even, then $y - x$ is also even, so $(y, x) \in R$ whenever $(x, y) \in R$.
  3. Transitive: If $x - y$ and $y - z$ are even, then $x - z = (x - y) + (y - z)$ is even.

Thus, $R$ is an equivalence relation.

Step 2: Equivalence classes

The equivalence class of an element $a$, denoted $\overline{a}$, is defined as: $\overline{a} = \{y \in \mathbb{Z} : (a, y) \in R\}.$ For $(a, y) \in R$, the condition $x - y$ is even implies that $a \equiv y \pmod{2}$. Hence, all integers in the same equivalence class as $a$ are congruent to $a$ modulo 2.

1. For $\overline{2}$:

   • The equivalence class $\overline{2}$ consists of all integers $y$ such that $2 - y$ is even.
   • This implies $y \equiv 2 \pmod{2}$, i.e., $y$ is an even integer.
   • Therefore, $\overline{2} = \{ \dots, -4, -2, 0, 2, 4, 6, \dots \}$.

2. For $\overline{-3}$:

   • The equivalence class $\overline{-3}$ consists of all integers $y$ such that $-3 - y$ is even.
   • This implies $y \equiv -3 \pmod{2}$, i.e., $y$ is an odd integer.
   • Therefore, $\overline{-3} = \{ \dots, -5, -3, -1, 1, 3, 5, \dots \}$.

Final Answer:

• $\overline{2} = \{ \dots, -4, -2, 0, 2, 4, 6, \dots \}$ (all even integers).
• $\overline{-3} = \{ \dots, -5, -3, -1, 1, 3, 5, \dots \}$ (all odd integers).

Would you like me to explain equivalence relations further or explore additional examples?

---

Relative Questions:

1. What are the key properties of equivalence relations?
2. How can equivalence classes partition the set of integers?
3. Can you derive equivalence classes for other values, like $\overline{0}$ or $\overline{1}$?
4. How would this equivalence relation change if the condition was $x - y \text{ is odd}$?
5. What are some real-world examples of equivalence relations?

Tip:

When determining equivalence classes, focus on identifying a defining property (like congruence modulo $n$) that all members share. This often simplifies the problem!