To prove that if $x = [a]$ and $y = [b]$, then the operation $x + y := [a + b]$ is well-defined on $\mathbb{Z}/n\mathbb{Z}$, let’s carefully go through each component of this statement.

Step-by-Step Proof

1. Understand the Definition of Equivalence Classes in $\mathbb{Z}/n\mathbb{Z}$:

   • The notation $\mathbb{Z}/n\mathbb{Z}$ represents the set of equivalence classes of integers under the equivalence relation $\sim_n$.
   • Two integers $a$ and $b$ are equivalent (written as $a \sim_n b$) if and only if their difference $a - b$ is divisible by $n$. This means there exists an integer $k$ such that: $a - b = kn.$

2. Define Addition in $\mathbb{Z}/n\mathbb{Z}$:

   • Suppose $x = [a]$ and $y = [b]$ are equivalence classes in $\mathbb{Z}/n\mathbb{Z}$.
   • We define the sum of $x$ and $y$ as: $x + y := [a + b].$
   • Our goal is to show that this definition does not depend on the representatives $a$ and $b$ chosen for $x$ and $y$, meaning that if we chose different representatives, the resulting equivalence class would still be the same.

3. Verify Well-Definedness:

   • Suppose $a'$ and $b'$ are other representatives of the classes $x$ and $y$, respectively, such that $a' \sim_n a$ and $b' \sim_n b$.
   • By the definition of $\sim_n$, we have: $a' = a + k_1 n \quad \text{and} \quad b' = b + k_2 n$ for some integers $k_1$ and $k_2$.
   • We want to show that: [ [a + b] = [a' + b']. ]
   • Substituting $a'$ and $b'$ in terms of $a$ and $b$, we get: $a' + b' = (a + k_1 n) + (b + k_2 n) = (a + b) + (k_1 + k_2) n.$
   • Therefore, $a' + b' \equiv a + b \pmod{n}$, which implies that $[a' + b'] = [a + b]$.

4. Conclusion:

   • This calculation shows that the equivalence class $[a + b]$ does not depend on the choice of representatives $a$ and $b$.
   • Hence, the operation $x + y := [a + b]$ is well-defined in $\mathbb{Z}/n\mathbb{Z}$.

Summary

We've shown that addition in $\mathbb{Z}/n\mathbb{Z}$ using the operation $x + y := [a + b]$ is well-defined, meaning that the sum of two equivalence classes does not depend on the choice of representatives. This allows us to treat $\mathbb{Z}/n\mathbb{Z}$ as a set with a consistent addition operation.

Would you like further details, or do you have any questions?

---

1. How can we define multiplication in $\mathbb{Z}/n\mathbb{Z}$?
2. What are some examples of equivalence classes in $\mathbb{Z}/n\mathbb{Z}$?
3. How does $\mathbb{Z}/n\mathbb{Z}$ relate to modular arithmetic?
4. Can you prove that $\mathbb{Z}/n\mathbb{Z}$ forms a group under addition?
5. What additional structure makes $\mathbb{Z}/n\mathbb{Z}$ a ring?

Tip: When proving that an operation is well-defined, always check if the operation yields consistent results independent of chosen representatives.