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Functional and not injective. For example, the red relation in the diagram is many-to-one, but the green, blue and black ones are not. Many-to-many [d] Not injective nor functional. For example, the black relation in the diagram is many-to-many, but the red, green and blue ones are not. Uniqueness and totality properties: A function [d]
A function that is not well defined is not the same as a function that is undefined. For example, if f ( x ) = 1 x {\displaystyle f(x)={\frac {1}{x}}} , then even though f ( 0 ) {\displaystyle f(0)} is undefined, this does not mean that the function is not well defined; rather, 0 is not in the domain of f {\displaystyle f} .
When all X i are the same set X, it is simpler to refer to R as an n-ary relation over X, called a homogeneous relation. Without this restriction, R is called a heterogeneous relation. When any of X i is empty, the defining Cartesian product is empty, and the only relation over such a sequence of domains is the empty relation R = ∅.
Then there is a unique function G such that for every x ∈ X, = (, | {:}). That is, if we want to construct a function G on X, we may define G(x) using the values of G(y) for y R x. As an example, consider the well-founded relation (N, S), where N is the set of all natural numbers, and S is the graph of the successor function x ↦ x+1. Then ...
For example, ≥ is a reflexive relation but > is not. Irreflexive (or strict) for all x ∈ X, not xRx. For example, > is an irreflexive relation, but ≥ is not. Coreflexive for all x, y ∈ X, if xRy then x = y. [7] For example, the relation over the integers in which each odd number is related to itself is a coreflexive relation.
A relation R is called intransitive if it is not transitive, that is, if xRy and yRz, but not xRz, for some x, y, z. In contrast, a relation R is called antitransitive if xRy and yRz always implies that xRz does not hold. For example, the relation defined by xRy if xy is an even number is intransitive, [13] but not antitransitive. [14] The ...
Although these terms are not further defined, Euclid uses them to construct more complex geometric concepts. [5] Whether a particular function or value is undefined, depends on the rules of the formal system in which it is used. For example, the imaginary number is undefined on the real number plane.
difunctional: the relation is the set {(,) =} for two partial functions,: and some indicator set right and left Euclidean : For a , b , c ∈ X {\displaystyle a,b,c\in X} , a R b {\displaystyle aRb} and a R c {\displaystyle aRc} implies b R c {\displaystyle bRc} and similarly for left Euclideanness b R a {\displaystyle bRa} and c R a ...