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A strict total order on a set is a strict ... Hence each subset of the real numbers is totally ordered, such as the natural numbers, integers, ...
Unlike the standard ordering ≤ of the natural numbers, the standard ordering ≤ of the integers is not a well ordering, since, for example, the set of negative integers does not contain a least element. The following binary relation R is an example of well ordering of the integers: x R y if and only if one of the following conditions holds ...
A partial order with this property is called a total order. These orders can also be called linear orders or chains. While many familiar orders are linear, the subset order on sets provides an example where this is not the case. Another example is given by the divisibility (or "is-a-factor-of") relation |.
A linear order or total order is defined by a set of elements and a comparison operation that gives an ordering to each pair of distinct elements and obeys the transitive law. [1] [2] The familiar numeric orderings on the integers, rational numbers, and real numbers are all examples of linear orders. [2]
A total order or linear order is a partial order under which every pair of elements is comparable, i.e. trichotomy holds. For example, the natural numbers with their standard order. A chain is a subset of a poset that is a totally ordered set. For example, {{}, {}, {,,}} is a chain.
A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering. The well-ordering theorem together with Zorn's lemma are the most important mathematical statements that are equivalent to the axiom of choice (often called AC, see also Axiom of choice § Equivalents).
The definition of total order appeared first historically and is a first-order axiomatization of the ordering as a binary predicate. Artin and Schreier gave the definition in terms of positive cone in 1926, which axiomatizes the subcollection of nonnegative elements.
By the definition of non-prime numbers, has factors ,, where , are integers greater than one and less than . Since a , b < n {\displaystyle a,b<n} , they are not in C {\displaystyle C} as n {\displaystyle n} is the smallest element of C {\displaystyle C} .