The relation $R$ on the set of integers $\mathbb{Z}$, defined by $aRb$ if and only if $a - b$ is divisible by 2, forms an equivalence relation. To verify that this relation is an equivalence relation, we need to check the three necessary properties: reflexivity, symmetry, and transitivity.

1. Reflexivity:

For all $a \in \mathbb{Z}$, $a - a = 0$, and 0 is divisible by 2. Therefore, $aRa$, so the relation is reflexive.

2. Symmetry:

If $aRb$, then $a - b = 2n$ for some integer $n$. This implies $b - a = -2n$, which is also divisible by 2, so $bRa$. Thus, the relation is symmetric.

3. Transitivity:

If $aRb$ (i.e., $a - b = 2n$ for some $n$) and $bRc$ (i.e., $b - c = 2m$ for some $m$), then: $a - c = (a - b) + (b - c) = 2n + 2m = 2(n + m).$ Since $a - c$ is divisible by 2, we have $aRc$. Therefore, the relation is transitive.

Since the relation satisfies reflexivity, symmetry, and transitivity, it is an equivalence relation.

Equivalence Classes:

The equivalence classes of this relation group integers based on whether their difference is divisible by 2. Specifically, the integers are classified based on their parity (whether they are even or odd).

• Class of even integers: All integers that are congruent to 0 modulo 2 (i.e., the even integers) form one equivalence class. This class can be represented by $[0]$, and it includes all even integers: $\{ \dots, -4, -2, 0, 2, 4, 6, \dots \}$.

• Class of odd integers: All integers that are congruent to 1 modulo 2 (i.e., the odd integers) form another equivalence class. This class can be represented by $[1]$, and it includes all odd integers: $\{ \dots, -3, -1, 1, 3, 5, \dots \}$.

Thus, the two equivalence classes under this relation are:

1. The set of all even integers: $[0] = \{ \dots, -4, -2, 0, 2, 4, \dots \}$
2. The set of all odd integers: $[1] = \{ \dots, -3, -1, 1, 3, 5, \dots \}$

Conclusion:

The equivalence classes of the relation $R$ partition the set of integers $\mathbb{Z}$ into two distinct classes:

• One class for the even integers.
• One class for the odd integers.

Would you like more details or have any questions?

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Here are 5 related questions to expand your understanding:

1. How would the equivalence classes change if the condition was divisibility by 3 instead of 2?
2. Can you describe an equivalence relation on $\mathbb{R}$ (real numbers) similar to this one?
3. How can you define an equivalence relation based on divisibility by any integer $n$?
4. How do equivalence relations relate to partitions of a set?
5. Can you provide an example of a relation that is not an equivalence relation?

Tip: Equivalence relations always partition a set into disjoint equivalence classes, and each element belongs to exactly one equivalence class.