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Order theory is a branch of mathematics that investigates the intuitive notion of order using binary ... A preorder is a relation that is reflexive and transitive ...
The resulting relation is reflexive since the preorder is reflexive; transitive by applying the transitivity of twice; and symmetric by definition. Using this relation, it is possible to construct a partial order on the quotient set of the equivalence, S / ∼ , {\displaystyle S/\sim ,} which is the set of all equivalence classes of ∼ ...
[8] [9] This definition is equivalent to a partial order on a setoid, where equality is taken to be a defined equivalence relation rather than set equality. [10] Wallis defines a more general notion of a partial order relation as any homogeneous relation that is transitive and antisymmetric. This includes both reflexive and irreflexive partial ...
An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations.
Order theory is a branch of mathematics that studies various kinds of objects (often binary relations) that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another. An alphabetical list of many notions of order theory can be found in the order theory glossary.
In social theory, reflexivity may occur when theories in a discipline should apply equally to the discipline itself; for example, in the case that the theories of knowledge construction in the field of sociology of scientific knowledge should apply equally to knowledge construction by sociology of scientific knowledge practitioners, or when the subject matter of a discipline should apply ...
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} :
Base.See continuous poset.; Binary relation.A binary relation over two sets is a subset of their Cartesian product.; Boolean algebra.A Boolean algebra is a distributive lattice with least element 0 and greatest element 1, in which every element x has a complement ¬x, such that x ∧ ¬x = 0 and x ∨ ¬x = 1.