Strict Ordering on Integers is Trichotomy
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Theorem
Let $\eqclass {a, b} {}$ and $\eqclass {c, d} {}$ be integers, as defined by the formal definition of integers.
Then exactly one of the following holds:
\(\ds \eqclass {a, b} {}\) | \(<\) | \(\ds \eqclass {c, d} {}\) | ||||||||||||
\(\ds \eqclass {a, b} {}\) | \(=\) | \(\ds \eqclass {c, d} {}\) | ||||||||||||
\(\ds \eqclass {a, b} {}\) | \(>\) | \(\ds \eqclass {c, d} {}\) |
That is, strict ordering is a trichotomy.
Proof
By the formal definition of integers, we have that $a, b, c, d, e, f$ are all natural numbers.
To eliminate confusion between integer ordering and the ordering on the natural numbers, let $a \preccurlyeq b$ denote that the natural number $a$ is less than or equal to the natural number $b$.
We have:
\(\ds \eqclass {a, b} {}\) | \(<\) | \(\ds \eqclass {c, d} {}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \eqclass {c, d} {}\) | \(\nless\) | \(\ds \eqclass {a, b} {}\) | Strict Ordering on Integers is Asymmetric |
Then:
\(\ds \eqclass {a, b} {}\) | \(<\) | \(\ds \eqclass {c, d} {}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds a + d\) | \(\prec\) | \(\ds b + c\) | Definition of Strict Ordering on Integers | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds a + d\) | \(\ne\) | \(\ds b + c\) | Definition of Ordering on Natural Numbers | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \eqclass {a, b} {}\) | \(\ne\) | \(\ds \eqclass {c, d} {}\) | Definition of Strict Ordering on Integers |
Similarly:
\(\ds \eqclass {a, b} {}\) | \(>\) | \(\ds \eqclass {c, d} {}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \eqclass {c, d} {}\) | \(\ngtr\) | \(\ds \eqclass {a, b} {}\) | Strict Ordering on Integers is Asymmetric |
Then:
\(\ds \eqclass {a, b} {}\) | \(>\) | \(\ds \eqclass {c, d} {}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds a + d\) | \(\succ\) | \(\ds b + c\) | Definition of Strict Ordering on Integers | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds a + d\) | \(\ne\) | \(\ds b + c\) | Definition of Ordering on Natural Numbers | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \eqclass {a, b} {}\) | \(\ne\) | \(\ds \eqclass {c, d} {}\) | Definition of Strict Ordering on Integers |
and:
\(\ds \eqclass {a, b} {}\) | \(=\) | \(\ds \eqclass {c, d} {}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds a + d\) | \(=\) | \(\ds b + c\) | Definition of Strict Ordering on Integers | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds a + d\) | \(\nprec\) | \(\ds b + c\) | Definition of Ordering on Natural Numbers | ||||||||||
\(\, \ds \land \, \) | \(\ds a + d\) | \(\nsucc\) | \(\ds b + c\) | Definition of Ordering on Natural Numbers | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \eqclass {a, b} {}\) | \(\nless\) | \(\ds \eqclass {c, d} {}\) | Definition of Strict Ordering on Integers | ||||||||||
\(\, \ds \land \, \) | \(\ds \eqclass {a, b} {}\) | \(\ngtr\) | \(\ds \eqclass {c, d} {}\) | Definition of Strict Ordering on Integers |
This demonstrates that $<$, $=$ and $>$ are mutually exclusive.
Now:
\(\ds \eqclass {a, b} {}\) | \(<\) | \(\ds \eqclass {c, d} {}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds a + d\) | \(\prec\) | \(\ds b + c\) | Definition of Strict Ordering on Integers | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds b + c\) | \(\preccurlyeq\) | \(\ds a + d\) | Definition of Ordering on Natural Numbers | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \eqclass {c, d} {}\) | \(\le\) | \(\ds \eqclass {a, b} {}\) | Definition of Strict Ordering on Integers |
Similarly:
\(\ds \eqclass {a, b} {}\) | \(>\) | \(\ds \eqclass {c, d} {}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds a + d\) | \(\succ\) | \(\ds b + c\) | Definition of Strict Ordering on Integers | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds b + c\) | \(\succcurlyeq\) | \(\ds a + d\) | Definition of Ordering on Natural Numbers | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \eqclass {c, d} {}\) | \(\ge\) | \(\ds \eqclass {a, b} {}\) | Definition of Strict Ordering on Integers |
and:
\(\ds \eqclass {a, b} {}\) | \(\ne\) | \(\ds \eqclass {c, d} {}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds a + d\) | \(\ne\) | \(\ds b + c\) | Definition of Strict Ordering on Integers | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds a + d\) | \(\prec\) | \(\ds b + c\) | Definition of Ordering on Natural Numbers | ||||||||||
\(\, \ds \lor \, \) | \(\ds a + d\) | \(\succ\) | \(\ds b + c\) | Definition of Ordering on Natural Numbers | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \eqclass {a, b} {}\) | \(<\) | \(\ds \eqclass {c, d} {}\) | Definition of Strict Ordering on Integers | ||||||||||
\(\, \ds \land \, \) | \(\ds \eqclass {a, b} {}\) | \(\>\) | \(\ds \eqclass {c, d} {}\) | Definition of Strict Ordering on Integers |
demonstrating that either $<$, $=$ or $>$ must hold.
$\blacksquare$
Sources
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 5$: The system of integers