$\forall x, y, z \in Z$ :

Closure

Integers are closed under addition and multiplication When you add or multiply an integer, you will get an integer $x + y \in Z$ and $x y \in Z$

Commutativity

$x + y = y + x$ and $x y = y x$

$x + y + z = (x + y) + z = x + (y + z)$ and $x yz = (x y) z = x (yz)$

Multiplication is distributive over addition (but not the other way around) $x (y + z) = x y + x z$

Exactly one of the following is true: $x = y$ or $x < y$ or $x > y$