To prove that the relation $\sim$ defined on $\mathbb{Z}$ by $a \sim b$ if $2a + b \equiv 0 \pmod{3}$ is an equivalence relation, we need to verify that it satisfies the three properties of an equivalence relation: reflexivity, symmetry, and transitivity.

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1. Reflexivity:

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

• For $a \sim a$, we need $2a + a \equiv 0 \pmod{3}$, which simplifies to: $3a \equiv 0 \pmod{3}.$
• Since $3a$ is always divisible by 3 for any integer $a$, the condition holds true. Thus, the relation is reflexive.

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2. Symmetry:

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

• Assume $a \sim b$, meaning $2a + b \equiv 0 \pmod{3}$. Rewriting, we get: $2a + b \equiv 0 \pmod{3} \implies b \equiv -2a \pmod{3}.$
• To check symmetry, we must show that $b \sim a$, i.e., $2b + a \equiv 0 \pmod{3}$. Substituting $b \equiv -2a \pmod{3}$ into $2b + a$, we have: $2(-2a) + a \equiv -4a + a \equiv -3a \equiv 0 \pmod{3}.$
• Thus, $b \sim a$, and the relation is symmetric.

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3. Transitivity:

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

• Assume $a \sim b$ and $b \sim c$. From $a \sim b$, we know: $2a + b \equiv 0 \pmod{3} \implies b \equiv -2a \pmod{3}.$ From $b \sim c$, we know: $2b + c \equiv 0 \pmod{3} \implies c \equiv -2b \pmod{3}.$
• Substituting $b \equiv -2a \pmod{3}$ into $c \equiv -2b \pmod{3}$, we get: $c \equiv -2(-2a) \equiv 4a \pmod{3}.$ Since $4a \equiv a \pmod{3}$ (as $4 \equiv 1 \pmod{3}$), we have $c \equiv a \pmod{3}$.
• To verify $a \sim c$, substitute back into $2a + c$: $2a + c \equiv 2a + a \equiv 3a \equiv 0 \pmod{3}.$
• Thus, $a \sim c$, and the relation is transitive.

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Conclusion:

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

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Do you have any questions or want further clarification? Here are some related questions to explore:

1. Can you provide examples of equivalence classes under this relation?
2. How do equivalence classes partition $\mathbb{Z}$?
3. Can this relation be generalized for $n$-modular arithmetic?
4. What is the significance of defining relations over modular arithmetic?
5. How would the proof change if the relation were defined differently, such as $a + b \equiv 0 \pmod{3}$?

Tip: Always check each property of an equivalence relation carefully, as skipping one can lead to incomplete proofs.