If R is the relation on the set of integers such that (a,b) E R when 3a+4b=7n for some integer n.
Prove that it is an equivalence relation

To prove that the relation \( R \) on the set of integers \( \mathbb{Z} \), defined by \((a, b) \in R \) if and only if \( 3a + 4b = 7n \) for some integer \( n \), is an equivalence relation, we need to show that it satisfies the three properties of an equivalence relation: **reflexivity**, **symmetry**, and **transitivity**.

### 1. Reflexivity
We need to show that for any integer \( a \), \((a, a) \in R\). 

For \((a, a) \) to be in \( R \), we need:
\[
3a + 4a = 7n
\]
Simplifying, we have:
\[
7a = 7n
\]
This simplifies further to:
\[
a = n
\]
Since \( n \) is an integer, there exists an integer \( n \) such that \( 3a + 4a = 7n \), specifically \( n = a \). Therefore, \((a, a) \in R \) for all \( a \in \mathbb{Z} \). Hence, the relation \( R \) is **reflexive**.

### 2. Symmetry
We need to show that if \( (a, b) \in R \), then \( (b, a) \in R \).

Assume \( (a, b) \in R \). This means that:
\[
3a + 4b = 7n \text{ for some integer } n.
\]
We need to show that there exists an integer \( m \) such that:
\[
3b + 4a = 7m.
\]

Notice that if \( 3a + 4b = 7n \), we can write it as:
\[
4b + 3a = 7n.
\]
This is identical in form to \( 3b + 4a = 7n \) with \( a \) and \( b \) swapped. Therefore, \( (b, a) \in R \), proving that \( R \) is **symmetric**.

### 3. Transitivity
We need to show that if \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \).

Assume \( (a, b) \in R \) and \( (b, c) \in R \). Then there exist integers \( n \) and \( m \) such that:
\[
3a + 4b = 7n \quad \text{and} \quad 3b + 4c = 7m.
\]
We want to show that \( (a, c) \in R \), i.e., that there exists an integer \( p \) such that:
\[
3a + 4c = 7p.
\]
To find such a relation, consider:
\[
3(3a + 4b) + 4(3b + 4c) = 3(7n) + 4(7m) = 21n + 28m.
\]
Simplifying, this gives:
\[
9a + 12b + 12b + 16c = 21n + 28m,
\]
\[
9a + 24b + 16c = 21n + 28m.
\]
This equation can be factored as:
\[
3(3a + 8b) + 4(4c) = 7(3n + 4m).
\]
This equation shows that the left-hand side is a multiple of 7, so:
\[
3a + 4c = 7p,
\]
where \( p = n + m \) is an integer. Therefore, \( (a, c) \in R \), and the relation \( R \) is **transitive**.

### Conclusion
Since \( R \) satisfies the properties of reflexivity, symmetry, and transitivity, \( R \) is an equivalence relation.

---

Would you like further details on any step? Here are some related questions:

1. How would this equivalence relation be represented as a partition of integers?
2. What is an example of an equivalence class under this relation?
3. Can we construct a different relation using a similar linear equation?
4. What are some real-life scenarios where equivalence relations are applied?
5. How does this relation differ from a partial order relation?

**Tip:** When proving properties of relations, always clearly define what each property means and methodically apply the definition to the given relation.

Learn how to prove that the relation R on the set of integers (a, b) where 3a + 4b = 7n for some integer n is an equivalence relation. This detailed mathematical proof covers reflexivity, symmetry, and transitivity of the relation R.

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