Let $\phi$ : $\mathbb{Z}$ x $\mathbb{Z}$ -> $\mathbb{Z}$ be the function defined by $\phi(a, b)$ = $a$ - $b$. Prove that $\phi$ is a homomorphism.
Definition of a homomorphism: Let $G, H$ be groups. A function $\phi$ : $G$ -> $H$ is a homomorphism if for all $a, b \in G$, $\phi(ab) = \phi(a)\phi(b)$. (sometimes say in this case that $\phi$ "respects the group structures.")
Hence, I need to show that for any $a, b \in$ $\mathbb{Z}$ x $\mathbb{Z}$, $\phi(ab) = \phi(a)\phi(b)$.
I do not know what $\mathbb{Z}$ x $\mathbb{Z}$ is. Likewise, I am unsure on how to show that for any $a, b \in$ $\mathbb{Z}$ x $\mathbb{Z}$, $\phi(ab) = \phi(a)\phi(b)$ as my function is defined by an ordered pair, namely $\phi(a, b)$ = $a$ - $b$. I would appreciate any feedback!