Search results
Results from the WOW.Com Content Network
A strict total order on a set is a strict partial order on in which any two distinct elements are comparable. That is, a strict total order is a binary relation < {\displaystyle <} on some set X {\displaystyle X} , which satisfies the following for all a , b {\displaystyle a,b} and c {\displaystyle c} in X {\displaystyle X} :
A strict weak order that is trichotomous is called a strict total order. [14] The total preorder which is the inverse of its complement is in this case a total order . For a strict weak order < {\displaystyle \,<\,} another associated reflexive relation is its reflexive closure , a (non-strict) partial order ≤ . {\displaystyle \,\leq .}
If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, then so too are its restrictions. However, the transitive closure of a restriction is a subset of the restriction of the ...
A law of trichotomy on some set X of numbers usually expresses that some tacitly given ordering relation on X is a trichotomous one. An example is the law "For arbitrary real numbers x and y, exactly one of x < y, y < x, or x = y applies"; some authors even fix y to be zero, [1] relying on the real number's additive linearly ordered group structure.
For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by in the "Symmetric" column and in the "Antisymmetric" column, respectively. All definitions tacitly require the homogeneous relation R {\displaystyle R} be transitive : for all a , b , c , {\displaystyle a,b,c,} if a R b {\displaystyle ...
In constructive mathematics, pseudo-order is a name given to certain binary relations appropriate for modeling continuous orderings.. In classical mathematics, its axioms constitute a formulation of a strict total order (also called linear order), which in that context can also be defined in other, equivalent ways.
"We don't want that," Jessica says in the video while sitting in her car. "You have 364/5 days to shop for us. Get us what we want. I sent Kirk my wishlist six weeks ago, so help me God.
For example, "divides " is a partial, but not a total order on natural numbers, " <" is a strict total order on , and "is parallel to " is an equivalence relation on the set of all lines in the Euclidean plane.