For each (non-strict) total order ≤ there is an associated asymmetric (hence irreflexive) relation <, called a strict total order, which can equivalently be defined in two ways:
- a < b if and only if a ≤ b and a ≠ b
- a < b if and only if not b ≤ a (i.e., < is the inverse of the complement of ≤)
Properties:
- The relation is transitive: a < b and b < c implies a < c.
- The relation is trichotomous: exactly one of a < b, b < a and a = b is true.
- The relation is a strict weak order, where the associated equivalence is equality.
We can work the other way and start by choosing < as a transitive trichotomous binary relation; then a total order ≤ can equivalently be defined in two ways:
- a ≤ b if and only if a < b or a = b
- a ≤ b if and only if not b < a
Two more associated orders are the complements ≥ and >, completing the quadruple {<, >, ≤, ≥}.
We can define or explain the way a set is totally ordered by any of these four relations; the notation implies whether we are talking about the non-strict or the strict total order.
Read more about this topic: Total Order
Famous quotes containing the words strict, total and/or order:
“The admission of Oriental immigrants who cannot be amalgamated with our people has been made the subject either of prohibitory clauses in our treaties and statutes or of strict administrative regulations secured by diplomatic negotiations. I sincerely hope that we may continue to minimize the evils likely to arise from such immigration without unnecessary friction and by mutual concessions between self-respecting governments.”
—William Howard Taft (1857–1930)
“Jarndyce and Jarndyce drones on. This scarecrow of a suit, has, in course of time, become so complicated that no man alive knows what it means. The parties to it understand it least; but it has been observed that no two Chancery lawyers can talk about it for five minutes, without coming to total disagreement as to all the premises.”
—Charles Dickens (1812–1870)
“There surely is a being who presides over the universe; and who, with infinite wisdom and power, has reduced the jarring elements into just order and proportion. Let speculative reasoners dispute, how far this beneficent being extends his care, and whether he prolongs our existence beyond the grave, in order to bestow on virtue its just reward, and render it fully triumphant.”
—David Hume (1711–1776)