[Math] Determine whether the given map $\phi$ is an isomorphism of the first binary structure with the second.

Question

Determine whether the given map $\phi$ is an isomorphism of the first binary structure with the second.

$\langle ℤ,+\rangle$ with $\langle ℤ,+\rangle$ where $\phi(n)=-n$ for $n$ an element of $ℤ$

I know in order to prove two sets are isomorphic they must be:
i) $1$-$1$ $f(\phi1)=f(\phi2)$ provided $\phi1=\phi2$
ii) onto: every element in the domain has a pre-image
iii) homomorphic $f(\phi1+\phi2)=f(\phi1)*f(\phi2)$

I am just confused how to prove this when the sets are the same. Wouldn't they automatically be isomorphic?

Answer 1

Condition (i) should be

If $\phi(x)=\phi(y)$ then $x=y$

Condition (iii) should be

$\phi(x+y)=\phi(x)+\phi(y)$

Now, can you prove that $-x=-y$ implies $x=y$?

Can you prove that $-(x+y)=(-x)+(-y)$?

What about surjectivity? Hint: it's easy.