To determine whether $R$ is an equivalence relation, we must verify if it satisfies the three properties of equivalence relations: reflexivity, symmetry, and transitivity.

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Definition of $R$:

A relation $R$ on $\mathbb{Z}$ (the set of integers) is defined as: $a R b \iff 3 \mid (5a - 8b),$ where $3 \mid x$ means that $x$ is divisible by 3.

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1. Reflexivity

A relation $R$ is reflexive if $a R a$ for all $a \in \mathbb{Z}$.

$a R a \iff 3 \mid (5a - 8a) = 3a.$

Since $3a$ is always divisible by 3 for any integer $a$, $R$ is reflexive.

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2. Symmetry

A relation $R$ is symmetric if $a R b \implies b R a$.

Suppose $a R b$, which means: $3 \mid (5a - 8b).$

Then, there exists an integer $k$ such that: $5a - 8b = 3k.$

We need to check whether $b R a$, which means: $3 \mid (5b - 8a).$

Compute $5b - 8a$ in terms of $5a - 8b$: $5b - 8a = -(5a - 8b).$

Since $3 \mid (5a - 8b)$, it follows that $3 \mid -(5a - 8b)$. Therefore, $3 \mid (5b - 8a)$, and $b R a$. Thus, $R$ is symmetric.

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3. Transitivity

A relation $R$ is transitive if $a R b$ and $b R c \implies a R c$.

Suppose $a R b$ and $b R c$. Then: $3 \mid (5a - 8b) \quad \text{and} \quad 3 \mid (5b - 8c).$

This means there exist integers $k_1$ and $k_2$ such that: $5a - 8b = 3k_1 \quad \text{and} \quad 5b - 8c = 3k_2.$

Add these two equations: $(5a - 8b) + (5b - 8c) = 3k_1 + 3k_2.$

Simplify: $5a - 8c = 3(k_1 + k_2).$

Since $3 \mid (5a - 8c)$, $a R c$. Thus, $R$ is transitive.

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Conclusion

The relation $R$ satisfies reflexivity, symmetry, and transitivity. Therefore, $R$ is an equivalence relation.

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Questions:

1. Can you provide an example to verify the transitivity of this relation using specific integers?
2. How would the properties change if the divisor were a number other than 3 (e.g., 4 or 5)?
3. Why does the symmetry property depend on the additive structure of integers?
4. How does divisibility ensure reflexivity for this specific relation?
5. Can $R$ partition $\mathbb{Z}$ into equivalence classes? If so, what do the equivalence classes look like?

Tip:

When checking equivalence relations, always try verifying the properties with small, concrete examples to build intuition.