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:
“Science asks no questions about the ontological pedigree or a priori character of a theory, but is content to judge it by its performance; and it is thus that a knowledge of nature, having all the certainty which the senses are competent to inspire, has been attaineda knowledge which maintains a strict neutrality toward all philosophical systems and concerns itself not with the genesis or a priori grounds of ideas.”
—Chauncey Wright (18301875)
“When I turned into a parent, I experienced a real and total personality change that slowly shifted back to the normal me, yet has not completely vanished. I believe the two levels are now superimposed, with an additional sprinkling of mortality intimations.”
—Sonia Taitz (20th century)
“Revolution is like the daughters of Pelias: it cuts humanity to pieces in order to rejuvenate it.”
—Georg Büchner (18131837)