Order Laws

There is a relation $<$ (less than) defined on the real numbers such that for each pair $a,b$ of real numbers, the statement ``$a<b$'' is either true or false, and such that the following conditions are satisfied:

Trichotomy law: For each pair $a,b$ of real numbers exactly one of the following statements is true:

$$a < b, \hspace{1in} a = b, \hspace{1in} b < a.$$ (C.49)

We say that a real number $p$ is positive if and only if $p>0,$ and we say that a real number $n$ is negative if and only if $n < 0.$ Thus as a special case of the trichotomy law we have:

If $a$ is a real number, then exactly one of the following statements is true:

$$\mbox{$a$\ is positive,} \hspace{1in} a=0, \hspace{1in} \mbox{$a$\ is negative}.$$ (C.50)

Sign laws: Let $a,b$ be real numbers. Then

$$\mbox{ if }\ a > 0 \mbox{ and }\ b > 0 \mbox{ then }\ ab > 0 \mbox{ and }\ a/b > 0,$$ (C.51)

$$\mbox{ if }\ a < 0 \mbox{ and }\ b > 0 \mbox{ then }\ ab < 0 \mbox{ and }\ a/b < 0,$$ (C.52)

$$\mbox{ if }\ a > 0 \mbox{ and }\ b < 0 \mbox{ then }\ ab < 0 \mbox{ and }\ a/b < 0,$$ (C.53)

$$\mbox{ if }\ a < 0 \mbox{ and }\ b < 0 \mbox{ then }\ ab > 0 \mbox{ and }\ a/b > 0,$$ (C.54)

$$\mbox{ if }\ a > 0 \mbox{ and }\ b > 0 \mbox{ then }\ a+b > 0,$$ (C.55)

$$\mbox{ if }\ a < 0 \mbox{ and }\ b < 0 \mbox{ then }\ a+b < 0.$$ (C.56)

Also,

$$\mbox{ $a$\ is positive if and only if $-a$\ is negative},$$ (C.57)

and

$$\mbox{ $a$\ is positive if and only if $a^{-1}$\ is positive.}$$ (C.58)

$$\mbox{ if }\ ab > 0 \mbox{ then }\ \mbox{either } (a > 0 \mbox{ and }\ b>0) \mbox{ or }\ (a<0 \mbox{ and }\ b<0).$$ (C.59)

$$\mbox{ if }\ ab < 0 \mbox{ then }\ \mbox{either }(a > 0 \mbox{ and }\ b < 0) \mbox{ or }\ (a < 0 \mbox{ and }\ b > 0).$$ (C.60)

$$\mbox{ if }\ a/b > 0 \mbox{ then }\ \mbox{either } (a > 0 \mbox{ and }\ b>0) \mbox{ or }\ (a<0 \mbox{ and }\ b<0).$$ (C.61)

$$\mbox{ if }\ a/b < 0 \mbox{ then }\ \mbox{either }(a > 0 \mbox{ and }\ b < 0) \mbox{ or }\ (a < 0 \mbox{ and }\ b > 0).$$ (C.62)

It follows immediately from the sign laws that for all real
numbers $a$

$$a^2 \geq 0 \mbox{ and }\ \mbox{ if }\ a \not= 0 \mbox{ then }\ a^2 > 0.$$ (C.63)

Here, as usual $a^2$ means $a \cdot a.$

Transitivity of $<$: Let $a,b,c$ be real numbers. Then

$$\mbox{ if }\ a < b \mbox{ and }\ b < c \mbox{ then }\ a < c.$$ (C.64)

We write $a \leq b$ as an abbreviation for ``either $a<b$ or $a = b$'', and we write $b>a$ to mean $a<b$. We also nest inequalities in the following way:

$$a < b \leq c = d < e$$

means

$$a < b \mbox{ and }\ b \leq c \mbox{ and }\ c = d \mbox{ and }\ d < e.$$

Addition of Inequalities: Let $a,b,c,d$ be real numbers. Then

$$\mbox{ if }\ a < b \mbox{ and }\ c < d \mbox{ then }\ a+c < b+d,$$ (C.65)

$$\mbox{ if }\ a \leq b \mbox{ and }\ c \leq d \mbox{ then }\ a+c \leq b+d,$$ (C.66)

$$\mbox{ if }\ a < b \mbox{ and }\ c \leq d \mbox{ then }\ a+c < b+d,$$ (C.67)

$$\mbox{ if }\ a < b \mbox{ then }\ a-c < b-c,$$ (C.68)

$$\mbox{ if }\ c < d \mbox{ then }\ -c > -d,$$ (C.69)

$$\mbox{ if }\ c < d \mbox{ then }\ a-c > a-d.$$ (C.70)

Multiplication of Inequalities: Let $a,b,c,d$ be real numbers.

$$\mbox{ if }\ a < b \mbox{ and }\ c > 0 \mbox{ then }\ ac < bc,$$ (C.71)

$$\mbox{ if }\ a < b \mbox{ and }\ c > 0 \mbox{ then }\ a/c < b/c ,$$ (C.72)

$$\mbox{ if }\ 0 < a < b \mbox{ and }\ 0 < c < d \mbox{ then }\ 0 < ac < bd,$$ (C.73)

$$\mbox{ if }\ a < b \mbox{ and }\ c < 0 \mbox{ then }\ ac > bc,$$ (C.74)

$$\mbox{ if }\ a < b \mbox{ and }\ c < 0 \mbox{ then }\ a/c > b/c ,$$ (C.75)

$$\mbox{ if }\ 0 < a \mbox{ and }\ a < b \mbox{ then }\ a^{-1} > b^{-1}.$$ (C.76)

Discreteness of Integers: If $n$ is an integer, then there are no integers between $n$ and $n+1$, i.e. there are no integers $k$ satisfying $n < k < n+1.$ A consequence of this is that

$$\mbox{If } k,n \mbox{ are integers, and } k < n+1, \mbox{ then } k \leq n.$$ (C.77)

If $x$ and $y$ are real numbers such that $y - x > 1$ then there is an integer $n$ such that

$$x < n < y.$$ (C.78)

Archimedean Property: Let $x$ be an arbitrary real number. Then

$$\mbox{there exists an integer $n$\ such that $n > x$.}$$ (C.79)