Let ∼ be a relation defined on Z by a∼b if 3a≡7b(mod 7). is it an equivalent relation? If so, find the distinct equivalent classes

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**.

---

### 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 \( k = 0, 1, \dots, 6 \).

---

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

---

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.

Let ∼ be a relation defined on Z by a∼b if 3a≡7b(mod 7). Is it an equivalence relation? If so, find the distinct equivalence classes.

Explore whether the relation defined by 3a ≡ 7b (mod 7) is an equivalence relation, with a detailed verification of reflexivity, symmetry, and transitivity. Discover the seven distinct equivalence classes in modular arithmetic. Suitable for high school students in Grades 10-12.

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