
# 2.3 Arithmetic of inequality

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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 $$a+c< b+d$$.