To determine if the given relation $\sim$ defined on $\mathbb{Z}$ by $3a \equiv 7b \pmod{7}$ is an equivalence relation, we must check the three properties: reflexivity, symmetry, and transitivity. If all hold, we will find the distinct equivalence classes.

1. Reflexivity

For any $a \in \mathbb{Z}$, we must show $a \sim a$, i.e., $3a \equiv 7a \pmod{7}$.
Since $7a \equiv 0 \pmod{7}$, it follows that: 3a \equiv 7a \pmod{7} \quad \text{(true for all a)}. Thus, the relation is reflexive.

2. Symmetry

For any $a, b \in \mathbb{Z}$, if $a \sim b$, then $b \sim a$.
Assume $3a \equiv 7b \pmod{7}$. Rearrange to: $3a - 7b \equiv 0 \pmod{7}.$ Since $7b \equiv 0 \pmod{7}$, this simplifies to: $3a \equiv 7b \implies 3b \equiv 7a \pmod{7}.$ Thus, $b \sim a$, and the relation is symmetric.

3. Transitivity

For any $a, b, c \in \mathbb{Z}$, if $a \sim b$ and $b \sim c$, then $a \sim c$.
Assume: $3a \equiv 7b \pmod{7} \quad \text{and} \quad 3b \equiv 7c \pmod{7}.$ From $3a \equiv 7b$, we know: $3a - 7b \equiv 0 \pmod{7}.$ From $3b \equiv 7c$, we know: $3b - 7c \equiv 0 \pmod{7}.$ Adding these two equations gives: $(3a - 7b) + (3b - 7c) \equiv 0 \pmod{7}.$ This simplifies to: $3a - 7c \equiv 0 \pmod{7}.$ Thus, $3a \equiv 7c \pmod{7}$, and $a \sim c$. The relation is transitive.

Conclusion

Since the relation is reflexive, symmetric, and transitive, it is an equivalence relation.

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Distinct Equivalence Classes

The equivalence class of $a \in \mathbb{Z}$ is: $[a] = \{ b \in \mathbb{Z} \mid 3a \equiv 7b \pmod{7} \}.$ Simplify $3a \equiv 7b \pmod{7}$. Since $7b \equiv 0 \pmod{7}$, this reduces to: $3a \equiv 0 \pmod{7}.$ Thus, the equivalence classes depend only on $3a \mod{7}$. Since $3$ and $7$ are coprime, there are $7$ residue classes: [ [0], [1], [2], [3], [4], [5], [6]. ]

Each equivalence class contains integers $a$ such that $a \equiv k \pmod{7}$, where $k = 0, 1, \dots, 6$.

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Summary

1. The relation $\sim$ is an equivalence relation.
2. There are 7 distinct equivalence classes: [ [0], [1], [2], [3], [4], [5], [6]. ]

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Do you want further clarification on the steps?
Here are related questions for deeper exploration:

1. How do modular arithmetic properties ensure reflexivity in relations like this?
2. Why do the equivalence classes correspond to the residue classes modulo 7?
3. How can the coprimality of $3$ and $7$ affect the distinct equivalence classes?
4. What happens if the modulus changes to a non-prime like 8 or 9?
5. How do equivalence relations apply to quotient groups in abstract algebra?

Tip: When dealing with equivalence relations in modular arithmetic, always reduce coefficients to simplify proofs.