
2.3 Arithmetic of inequality

Definition

Let $$a, b\in \mathbb{Z}$$. Then

1. $$a< b$$ provided $$b=a + k$$, for some $$k \in \mathbb{Z_+}$$.
2. $$a> b$$ provided $$a=b + h$$, for some $$h \in \mathbb{Z_+}$$.

Let $$a, b\in \mathbb{Z}$$.

1.  If  $$a< b$$  then  $$a+c< b+c$$, $$\forall c \in \mathbb{Z}$$.
2.  If  $$a< b$$  then  $$ac< bc$$,$$\forall c \in \mathbb{Z_+}$$.
3.  If  $$a< b$$  and   $$c< d$$  then   $$a+c< b+d$$.