Show that (𝑥 + 𝑦) + 𝑧 = 𝑥 + (𝑦 + 𝑧) for all 𝑥, 𝑦, 𝑧 ∈ 𝐅𝑛

To prove that \((x + y) + z = x + (y + z)\) for all \(x, y, z \in \mathbb{F}_n\), we need to show that the operation of addition in the finite field \(\mathbb{F}_n\) is associative.

### Step 1: Define the Elements
Let \(x, y, z\) be elements of the finite field \(\mathbb{F}_n\). In \(\mathbb{F}_n\), addition is defined modulo \(n\). Therefore, for any two elements \(a, b \in \mathbb{F}_n\), the addition \(a + b\) is given by the usual integer addition followed by taking the result modulo \(n\).

### Step 2: Apply Addition Modulo \(n\)
We want to prove the associative property of addition:

\[
(x + y) + z = x + (y + z)
\]

Let's consider both sides of this equation.

#### Left-Hand Side:
\[
(x + y) + z = [(x + y) \mod n] + z
\]
Now, add \(z\) to the result:
\[
= [(x + y) \mod n + z] \mod n
\]

#### Right-Hand Side:
\[
x + (y + z) = x + [(y + z) \mod n]
\]
Now, add \(x\) to the result:
\[
= [x + (y + z) \mod n] \mod n
\]

### Step 3: Use Properties of Modular Arithmetic
In modular arithmetic, addition is associative. Therefore:
\[
[(x + y) \mod n + z] \mod n = [x + (y + z) \mod n] \mod n
\]
This simplifies to:
\[
[(x + y + z) \mod n] = [(x + y + z) \mod n]
\]

### Step 4: Conclusion
Since both sides are equivalent, we have shown that:
\[
(x + y) + z = x + (y + z)
\]
for all \(x, y, z \in \mathbb{F}_n\).

### Summary
The operation of addition in the finite field \(\mathbb{F}_n\) is associative, and thus \((x + y) + z = x + (y + z)\) holds true for any elements \(x, y, z\) in \(\mathbb{F}_n\).

Do you want more details or have any questions?

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**Related Questions:**
1. What is the definition of a finite field \(\mathbb{F}_n\)?
2. How does modular arithmetic work in \(\mathbb{F}_n\)?
3. What are the properties of addition in a finite field?
4. How do you prove the commutative property of addition in \(\mathbb{F}_n\)?
5. Can you demonstrate the distributive property in \(\mathbb{F}_n\)?

**Tip:** Associativity is a fundamental property of binary operations, and it's essential for the structure of fields and rings in abstract algebra.

Learn how to prove the associativity of addition in finite fields using modular arithmetic. This problem explores the fundamental property of associativity in abstract algebra, specifically in the context of finite fields. Suitable for undergraduate students studying abstract algebra or advanced mathematics.

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