The Order Axioms

There is a relation called $<$ that's defined for all real numbers — that is, given any numbers $x, y\in\mathbb{R}$, then $x < y$ is either true or false. This relation satisfies the following properties:

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  • (a) Trichotomy ("tri-" = three; "-chotomy" = cutting): For any $x, y\in\mathbb{R}$, exactly one of the following three possibilities is true: $x < y$, $y<x$, or $x=y$. Taking $y=0$, we can define two new words:
    • $0 < x$, in which case we say that $x$ is positive;
    • $x < 0$, in which case we say that $x$ is negative; or
    • $x = 0$.
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  • (b) Transitivity: If $x < y$ and $y < z$, then $x < z$.
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  • (c) Addition preserves order: If $x < y$, then $x+z < y+z$ for any real number $z$.
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  • (d) Multiplication preserves positivity: If $0<x$ and $0 < y$, then $0 < x\cdot y$. (So we say that the positive numbers are closed under multiplication.)
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I've been careful to write all of the things above using only the $<$ sign, but it will also be convenient for us to use the $>$ sign in the usual way: $x > y$ means the same thing as $y < x$. Same with $\leq$, which means "either $x < y$ or $x = y$", and $\geq$, which means something similar.

A note on the trichotomy axiom: The fact that exactly one of the three possibilities is true is more powerful than you think. For instance, if you know that $a > 0$, then that means that $a\neq 0$ (and that $a \not\lt 0$). In words, if $a$ is positive, then $a$ is not zero, and $a$ is not negative.