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In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation ≤ {\displaystyle \leq } 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 relation that is functional and total. For example, the red and green relations in the diagram are functions, but the blue and black ones are not. An injection [d] A function that is injective. For example, the green relation in the diagram is an injection, but the red, blue and black ones are not. A surjection [d] A function that is surjective.
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that".
A linear extension is an extension that is also a linear (that is, total) order. As a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order. Every partial order can be extended to a total order (order-extension principle). [16]
Order, an academic journal on order theory; Dense order, a total order wherein between any unequal pair of elements there is always an intervening element in the order; Glossary of order theory; Lexicographical order, an ordering method on sequences analogous to alphabetical order on words; List of order topics, list of order theory topics
In mathematics, a recurrence relation is an equation according to which the th term of a sequence of numbers is equal to some combination of the previous terms. Often, only previous terms of the sequence appear in the equation, for a parameter that is independent of ; this number is called the order of the relation.
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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.
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