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 $a +_5 e = a$