The problem states:

Let $G$ be the set defined by $G = \{(a, b) \mid a, b \in \mathbb{R}, ab \geq 0\}.$ We are also given an operation $*$ defined on $G$ by: $(a, b) * (c, d) = (a + c, b + d).$ The question is whether $G$ forms a group under this operation $*$, and we are asked to prove it.

Solution Outline

To determine if $G$ is a group under the operation $*$, we need to verify the four group axioms: closure, associativity, identity, and invertibility.

Step 1: Closure

To verify closure, we need to check that if $(a, b) \in G$ and $(c, d) \in G$, then $(a + c, b + d) \in G$.

Since $(a, b) \in G$, we know $ab \geq 0$ and similarly $cd \geq 0$ for $(c, d) \in G$. Now we examine the product of the components of $(a + c, b + d)$: $(a + c)(b + d) = ab + ad + bc + cd.$ Since $ab \geq 0$ and $cd \geq 0$, we need to check if this sum remains non-negative.

Step 2: Associativity

The operation $*$ must be associative, meaning $(a, b) * ((c, d) * (e, f)) = ((a, b) * (c, d)) * (e, f)$ for all $(a, b), (c, d), (e, f) \in G$.

Step 3: Identity Element

We need to find an element $(e, f) \in G$ such that $(a, b) * (e, f) = (a, b)$ for any $(a, b) \in G$.

Step 4: Inverse Element

For each $(a, b) \in G$, there should exist an element $(c, d) \in G$ such that $(a, b) * (c, d) = (0, 0)$, where $(0, 0)$ is the identity element.

Let's go through these steps in detail:

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