There is a relation (less than) defined on the real
numbers such that for each pair of real numbers, the statement ``''
is either true or false, and such that the following conditions are satisfied:
Trichotomy law: For each pair of real numbers exactly
one of the following statements is true:
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We say that a real number is positive if and only if and we say that a real number is negative if and only if Thus as a special case of the trichotomy law we have:
If is a real number, then exactly one of the following
statements is true:
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Sign laws: Let be real numbers. Then
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Transitivity of :
Let be real numbers. Then
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We write as an abbreviation for ``either or '',
and we write to mean . We also nest inequalities in the
following way:
Addition of Inequalities: Let be real numbers. Then
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Multiplication of Inequalities: Let be real numbers.
Discreteness of Integers:
If is an integer, then there are no integers between and ,
i.e. there are no integers satisfying A consequence
of this is that
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Archimedean Property: Let be an arbitrary real number.
Then