To prove that \([ \mathbb{Z}_5, +_5 ]\) is an Abelian group, we need to verify that it satisfies the group axioms for being a group, and then confirm the additional property that it is Abelian (i.e., commutative under addition). Let's go through each step in detail.

### Step 1: **Set Definition**
\[
\mathbb{Z}_5 = \{ 0, 1, 2, 3, 4 \}
\]
This set consists of the integers modulo 5, i.e., the remainders when dividing by 5. The operation \( +_5 \) denotes addition modulo 5.

### Step 2: **Group Axioms**

1. **Closure:**
   Closure means that the sum of any two elements in \(\mathbb{Z}_5\), under \(+_5\), is also an element of \(\mathbb{Z}_5\).
   
   Let \(a, b \in \mathbb{Z}_5\). Then,
   \[
   a +_5 b = (a + b) \mod 5
   \]
   The result of this operation is always an element of \(\mathbb{Z}_5\), because after taking the modulus, the result will be one of \(\{ 0, 1, 2, 3, 4 \}\).

   Thus, closure holds.

2. **Associativity:**
   We need to check if \((a +_5 b) +_5 c = a +_5 (b +_5 c)\) for all \(a, b, c \in \mathbb{Z}_5\).

   Since \(+_5\) is addition modulo 5, and ordinary addition is associative, it follows that:
   \[
   ((a + b) \mod 5) + c = (a + (b + c)) \mod 5
   \]
   Therefore, associativity holds.

3. **Identity Element:**
   The identity element \(e\) in a group is an element such that \(a +_5 e = a\)

This detailed solution demonstrates that [Z5, +5] is an abelian group. It covers core group theory concepts, including closure, associativity, identity, inverses, and commutativity, using modulo arithmetic in Z5. Ideal for university-level abstract algebra students.

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