Let ∼ be a relation on the set N × N, where (m, n) ∼ (p, q), if m + q = n + p. Show that ∼ is an equivalence relation on the set N × N.

To prove that the given relation \( \sim \) on \( \mathbb{N} \times \mathbb{N} \) defined by \( (m, n) \sim (p, q) \iff m + q = n + p \) is an equivalence relation, we must show that \( \sim \) satisfies the three properties of an equivalence relation: **reflexivity, symmetry, and transitivity**.

### 1. **Reflexivity**:
We need to show that \( (m, n) \sim (m, n) \) for all \( (m, n) \in \mathbb{N} \times \mathbb{N} \).

\[
(m, n) \sim (m, n) \iff m + n = n + m.
\]

Since addition is commutative, \( m + n = n + m \) is always true. Hence, \( \sim \) is reflexive.

---

### 2. **Symmetry**:
We need to show that if \( (m, n) \sim (p, q) \), then \( (p, q) \sim (m, n) \).

Assume \( (m, n) \sim (p, q) \), which means \( m + q = n + p \). 

Switching the pairs, we must show \( (p, q) \sim (m, n) \), i.e., \( p + n = q + m \).

Since \( m + q = n + p \), this equality can be rewritten as \( p + n = q + m \) (simply rearranging terms). Hence, \( \sim \) is symmetric.

---

### 3. **Transitivity**:
We need to show that if \( (m, n) \sim (p, q) \) and \( (p, q) \sim (r, s) \), then \( (m, n) \sim (r, s) \).

From \( (m, n) \sim (p, q) \), we have:
\[
m + q = n + p.
\]

From \( (p, q) \sim (r, s) \), we have:
\[
p + s = q + r.
\]

We must show that \( (m, n) \sim (r, s) \), i.e., \( m + s = n + r \).

Add the two given equalities:
\[
(m + q) + (p + s) = (n + p) + (q + r).
\]

Simplify:
\[
m + s + q + p = n + r + p + q.
\]

Cancel \( q + p \) from both sides:
\[
m + s = n + r.
\]

Thus, \( (m, n) \sim (r, s) \), and \( \sim \) is transitive.

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### Conclusion:
Since \( \sim \) satisfies reflexivity, symmetry, and transitivity, it is an equivalence relation on \( \mathbb{N} \times \mathbb{N} \).

---

Would you like me to expand this explanation further, or clarify any steps? Here are some related questions:

1. What is an equivalence class under this relation, and how is it structured?
2. How many equivalence classes exist for this relation on \( \mathbb{N} \times \mathbb{N} \)?
3. Can you construct a specific example of equivalence classes under this relation?
4. What other properties of equivalence relations are useful for understanding set partitions?
5. How can we represent \( \mathbb{N} \times \mathbb{N} \) geometrically under this equivalence relation?

**Tip**: Remember that equivalence relations always partition a set into distinct equivalence classes!

Explore how to prove that the relation ∼ on N × N, defined by (m, n) ∼ (p, q) if m + q = n + p, is an equivalence relation. Understand reflexivity, symmetry, and transitivity with detailed mathematical reasoning and examples. Perfect for undergraduate mathematics students.

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