Definition:Ordering/Notation
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Definition
Symbols used to denote a general ordering relation are usually variants on $\preceq$, $\le$ and so on.
On $\mathsf{Pr} \infty \mathsf{fWiki}$, to denote a general ordering relation it is recommended to use $\preceq$ and its variants:
- $\preccurlyeq$
- $\curlyeqprec$
To denote the conventional ordering relation in the context of numbers, the symbol $\le$ is to be used, or its variants:
- $\leqslant$
- $\leqq$
- $\eqslantless$
The symbol $\subseteq$ is universally reserved for the subset relation.
\(\ds a\) | \(\preceq\) | \(\ds b\) | can be read as: | \(\quad\) $a$ precedes, or is the same as, $b$ | ||||||||||
\(\ds a\) | \(\preceq\) | \(\ds b\) | can be read as: | \(\quad\) $b$ succeeds, or is the same as, $a$ |
If, for two elements $a, b \in S$, it is not the case that $a \preceq b$, then the symbols $a \npreceq b$ and $b \nsucceq a$ can be used.
When the symbols $\le$ and its variants are used, it is common to interpret them as follows:
\(\ds a\) | \(\le\) | \(\ds b\) | can be read as: | \(\quad\) $a$ is less than, or is the same as, $b$ | ||||||||||
\(\ds a\) | \(\le\) | \(\ds b\) | can be read as: | \(\quad\) $b$ is greater than, or is the same as, $a$ |
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering