In mathematics, a binary relation on a set is reflexive if it relates every element of to itself.
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.
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Relations are ways in which several entities stand to each other. They usually connect distinct entities but some associate an entity with itself. The adicity of a relation is the number of entities it connects. The direction of a relation is the order in which the elements are related to each other. The converse of a relation carries the same information and has the opposite direction, like the contrast between "two is less than five" and "five is greater than two". Both relations and properties express features in reality with a key difference being that relations apply to several entities while properties belong to a single entity.
Many types of relations are discussed in the academic literature. Internal relations, like resemblance, depend only on the monadic properties of the relata. They contrast with external relations, like spatial relations, which express characteristics that go beyond what their relata are like. Formal relations, like identity, involve abstract and topic-neutral ideas while material relations, like loving, have concrete and substantial contents. Logical relations are relations between propositions while causal relations connect concrete events. Symmetric, transitive, and reflexive relations are distinguished by their structural features.
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word partial is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable.
Formally, a partial order is a homogeneous binary relation that is reflexive, antisymmetric, and transitive. A partially ordered set (poset for short) is an ordered pair consisting of a set (called the ground set of ) and a partial order on . When the meaning is clear from context and there is no ambiguity about the partial order, the set itself is sometimes called a poset.
In mathematics, equality is a relationship between two quantities or expressions, stating that they have the same value, or represent the same mathematical object. Equality between A and B is denoted with an equals sign as A = B, and read "A equals B". A written expression of equality is called an equation or identity depending on the context. Two objects that are not equal are said to be distinct.
Equality is often considered a primitive notion, meaning it is not formally defined, but rather informally said to be "a relation each thing bears to itself and nothing else". This characterization is notably circular ("nothing else"), reflecting a general conceptual difficulty in fully characterizing the concept. Basic properties about equality like reflexivity, symmetry, and transitivity have been understood intuitively since at least the ancient Greeks, but were not symbolically stated as general properties of relations until the late 19th century by Giuseppe Peano. Other properties like substitution and function application weren't formally stated until the development of symbolic logic.
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is numerical equality. Any number is equal to itself (reflexive). If , then (symmetric). If and , then (transitive).
Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class.
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 on some set , which satisfies the following for all and in :
Requirements 1. to 3. just make up the definition of a partial order.Reflexivity (1.) already follows from strong connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders.Total orders are sometimes also called simple, connex, or full orders.
