Definition:Strict Ordering/Asymmetric and Transitive
Definition
Let $\RR$ be a relation on a set $S$.
Then $\RR$ is a strict ordering (on $S$) if and only if $\RR$ satisfies the strict ordering axioms:
\((1)\) | $:$ | Asymmetry | \(\ds \forall a, b \in S:\) | \(\ds a \mathrel \RR b \) | \(\ds \implies \) | \(\ds \neg \paren {b \mathrel \RR a} \) | |||
\((2)\) | $:$ | Transitivity | \(\ds \forall a, b, c \in S:\) | \(\ds \paren {a \mathrel \RR b} \land \paren {b \mathrel \RR c} \) | \(\ds \implies \) | \(\ds a \mathrel \RR c \) |
Notation
Symbols used to denote a general strict ordering are usually variants on $\prec$, $<$ and so on.
On $\mathsf{Pr} \infty \mathsf{fWiki}$, to denote a general strict ordering it is recommended to use $\prec$.
To denote the conventional strict ordering in the context of numbers, the symbol $<$ is to be used.
The symbol $\subset$ is deprecated on $\mathsf{Pr} \infty \mathsf{fWiki}$, as it is ambiguous in the literature, and this can be a cause of confusion and conflict.
Hence the symbols $\subsetneq$ and $\subsetneqq$ are used for the (proper) subset relation.
\(\ds a\) | \(\prec\) | \(\ds b\) | can be read as: | \(\quad\) $a$ (strictly) precedes $b$ | ||||||||||
\(\ds a\) | \(\prec\) | \(\ds b\) | can also be read as: | \(\quad\) $b$ (strictly) succeeds $a$ |
If, for two elements $a, b \in S$, it is not the case that $a \prec b$, then the symbols $a \nprec b$ and $b \nsucc a$ can be used.
Also see
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 14$: Order
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets