Definition:Usual Ordering

From ProofWiki
Jump to navigation Jump to search

Definition

Let $X$ be one of the sets of numbers: $\N$, $\Z$, $\Q$, $\R$.

The usual ordering on $X$ is the conventional counting and measuring order on $X$ that is learned when one is initially introduced to numbers.


Natural Numbers

Let $\N$ denote the natural numbers.

The ordering on $\N$ is the relation $\le$ everyone is familiar with.

For example, we use it when we say:

James has $6$ apples, which is more than Mary, who has $4$.

which can be symbolised as:

$6 \ge 4$


Integers

The integers are ordered on the relation $\le$ as follows:

$\forall x, y \in \Z: x \le y$

if and only if:

$\exists c \in P: x + c = y$

where $P$ is the set of positive integers.


That is, $x$ is less than or equal to $y$ if and only if $y - x$ is non-negative.



This article is complete as far as it goes, but it could do with expansion.
In particular: This needs to be formalised. The page is included as it is one of the most wanted redline links.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Expand}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.


Also known as

The usual ordering is also known as the natural ordering.


Also see

  • Results about the usual ordering can be found here.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Usual_Ordering&oldid=705609"