Prove the following properties of $\mathbb{Z}$:
(a) (Trichotomy) For any $x, y ∈ \mathbb{Z}$, precisely one of $x < y$, $x = y$, $x > y$ is true.
(b) (Multiplicative ordering) For $x, y,z ∈ \mathbb{Z}$, if $x < y$ and $z > 0$ then $x ⊗ z < y ⊗ z$.
I have done part (a) as follows: Since $x, y ∈ \mathbb{Z}$, $x-y ∈ \mathbb{Z}$ by the Closure Property of Integers (Subtraction). As $x-y<0$ or $x-y>0$ or $x-y=0$, it follows that either $x<y$ or $x>y$ or $x=y$ respectively.
Another way I was thinking of doing part (a) is:
$∀x,y∈\mathbb{Z}:x\le y∨y\ge x$
$∀x.y∈\mathbb{Z}:x\le y∨x\ge y$
$∀x,y∈\mathbb{Z}:(x=y∨x<y)∨(x=y∨x>y) $
$∀x,y∈\mathbb{Z}:x<y∨x=y∨x>y$
Are any of the above methods sufficient to solve part (a)? Also, is this related to part (b) since I am unable to progress in (b).