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In order of increasing strength, i.e., decreasing sets of pairs, three of the possible partial orders on the Cartesian product of two partially ordered sets are (see Fig. 4): the lexicographical order: (a, b) ≤ (c, d) if a < c or (a = c and b ≤ d); the product order: (a, b) ≤ (c, d) if a ≤ c and b ≤ d;
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
In set theory, a branch of mathematics, a rank-into-rank embedding is a large cardinal property defined by one of the following four axioms given in order of increasing consistency strength. (A set of rank < λ {\displaystyle <\lambda } is one of the elements of the set V λ {\displaystyle V_{\lambda }} of the von Neumann hierarchy .)
A monotonically non-increasing function Figure 3. A function that is not monotonic. In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. [1] [2] [3] This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
As an example, "is less than" is a relation on the set of natural numbers; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3), and likewise between 3 and 4 (denoted as 3 < 4), but not between the values 3 and 1 nor between 4 and 4, that is, 3 < 1 and 4 < 4 both evaluate to false. As another example, "is sister of " is a ...
In an ordered set, one can define many types of special subsets based on the given order. A simple example are upper sets; i.e. sets that contain all elements that are above them in the order. Formally, the upper closure of a set S in a poset P is given by the set {x in P | there is some y in S with y ≤ x}. A set that is equal to its upper ...
For a finite set S, there is a bijection between the set of possible total orderings of the elements and the set of bijections from S to S. That is to say, the number of permutations of elements of S is the same as the number of total orderings of that set—namely, n!.
The unique order on the empty set, ∅, is a total order. Any set of cardinal numbers or ordinal numbers (more strongly, these are well-orders). If X is any set and f an injective function from X to a totally ordered set then f induces a total ordering on X by setting x 1 ≤ x 2 if and only if f(x 1) ≤ f(x 2).