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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]
The name is somewhat of a misnomer, since technically it is a class of binary relations, not functions, as the following formal definition explains: A binary relation P( x , y ), where y is at most polynomially longer than x , is in FNP if and only if there is a deterministic polynomial time algorithm that can determine whether P( x , y ) holds ...
For example, the green relation in the diagram is an injection, but the red one is not; the black and the blue relation is not even a function. A surjection: a function that is surjective. For example, the green relation in the diagram is a surjection, but the red one is not. A bijection: a function that is injective and surjective.
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 ...
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.
In constructive mathematics, "not empty" and "inhabited" are not equivalent: every inhabited set is not empty but the converse is not always guaranteed; that is, in constructive mathematics, a set that is not empty (where by definition, "is empty" means that the statement () is true) might not have an inhabitant (which is an such that ).
A common type of implicit function is an inverse function.Not all functions have a unique inverse function. If g is a function of x that has a unique inverse, then the inverse function of g, called g −1, is the unique function giving a solution of the equation
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 ...