Transitive relation in the context of Transitive closure

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, a binary relation ${\displaystyle R}$ on a set ${\displaystyle X}$ is reflexive if it relates every element of ${\displaystyle X}$ 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.

In set theory a serial relation is a homogeneous relation expressing the connection of an element of a sequence to the following element. The successor function used by Peano to define natural numbers is the prototype for a serial relation.

Bertrand Russell used serial relations in The Principles of Mathematics (1903) as he explored the foundations of order theory and its applications. The term serial relation was also used by B. A. Bernstein for an article showing that particular common axioms in order theory are nearly incompatible: connectedness, irreflexivity, and transitivity.

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 ${\displaystyle P=(X,\leq )}$ consisting of a set ${\displaystyle X}$ (called the ground set of ${\displaystyle P}$) and a partial order ${\displaystyle \leq }$ on ${\displaystyle X}$. When the meaning is clear from context and there is no ambiguity about the partial order, the set ${\displaystyle X}$ 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.

Interpretant is a subject / sign that refers to the same object as another sign, transitively.

In economics, and in other social sciences, preference refers to an order by which an agent, while in search of an "optimal choice", ranks alternatives based on their respective utility. Preferences are evaluations that concern matters of value, in relation to practical reasoning. Individual preferences are determined by taste, need, ..., as opposed to price, availability or personal income. Classical economics assumes that people act in their best (rational) interest. In this context, rationality would dictate that, when given a choice, an individual will select an option that maximizes their self-interest. But preferences are not always transitive, both because real humans are far from always being rational and because in some situations preferences can form cycles, in which case there exists no well-defined optimal choice. An example of this is Efron dice.

The concept of preference plays a key role in many disciplines, including moral philosophy and decision theory. The logical properties that preferences possess also have major effects on rational choice theory, which in turn affects all modern economic topics.

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 ${\displaystyle a}$ is equal to itself (reflexive). If ${\displaystyle a=b}$, then ${\displaystyle b=a}$ (symmetric). If ${\displaystyle a=b}$ and ${\displaystyle b=c}$, then ${\displaystyle a=c}$ (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 ${\displaystyle \leq }$ on some set ${\displaystyle X}$, which satisfies the following for all ${\displaystyle a,b}$ and ${\displaystyle c}$ in ${\displaystyle X}$:

1. ${\displaystyle a\leq a}$ (reflexive).
2. If ${\displaystyle a\leq b}$ and ${\displaystyle b\leq c}$ then ${\displaystyle a\leq c}$ (transitive).
3. If ${\displaystyle a\leq b}$ and ${\displaystyle b\leq a}$ then ${\displaystyle a=b}$ (antisymmetric).
4. ${\displaystyle a\leq b}$ or ${\displaystyle b\leq a}$ (strongly connected, formerly called totality).

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.

It is possible to find sets of dice with the even stronger property that, for each die in the set, there is another die that rolls a higher number than it more than half the time. This is different in that instead of only "${\displaystyle X_{1}}$ does not normally beat ${\displaystyle X_{n}}$" it is now "${\displaystyle X_{n}}$ normally beats ${\displaystyle X_{1}}$". Using such a set of dice, one can invent games which are biased in ways that people unused to intransitive dice might not expect (see example).

In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with each other. Weak orders are a generalization of totally ordered sets (rankings without ties) and are in turn generalized by (strictly) partially ordered sets and preorders.

There are several common ways of formalizing weak orderings, that are different from each other but cryptomorphic (interconvertable with no loss of information): they may be axiomatized as strict weak orderings (strictly partially ordered sets in which incomparability is a transitive relation), as total preorders (transitive binary relations in which at least one of the two possible relations exists between every pair of elements), or as ordered partitions (partitions of the elements into disjoint subsets, together with a total order on the subsets). In many cases another representation called a preferential arrangement based on a utility function is also possible.