Answer to Question #211721 in Abstract Algebra for Faizee

Question

Prove that (Q, +) is an abelian group under ordinary addition.

Answer 1

Let Q be the set of rational numbers.

Rational Addition is Closed

The operation of addition on the set of rational numbers Q is well-defined and closed:

∀ (x , y) ∈ Q : x + y ∈ Q

Rational Addition is Associative

The operation of addition on the set of rational numbers Q is associative:

∀ (x , y , z) ∈ Q : x + (y + z) = (x + y) + z

Rational Addition Identity is Zero

The identity of rational number addition is 0:

∃ 0 ∈ Q : ∀ a ∈ Q : a + 0 = a = 0 + a

Inverse for Rational Addition

Each element x of the set of rational numbers Q has an inverse element −x under the operation of rational number addition:

∀ x ∈ Q : ∃ − x ∈ Q : x + (−x) = 0 = (−x) + x

Rational Addition is Commutative

The operation of addition on the set of rational numbers Q is commutative:

∀ (x , y) ∈ Q : x + y = y + x

Hence, the structure (Q,+) is an abelian group under ordinary addition.