Search results
Results from the WOW.Com Content Network
For example, the green and blue relations in the diagram are injective, but the red one is not (as it relates both −1 and 1 to 1), nor is the black one (as it relates both −1 and 1 to 0). Functional [ 15 ] [ 16 ] [ 17 ] [ d ] (also called right-unique , [ 14 ] right-definite [ 18 ] or univalent [ 9 ] )
Dirac delta function: everywhere zero except for x = 0; total integral is 1. Not a function but a distribution, but sometimes informally referred to as a function, particularly by physicists and engineers. Dirichlet function: is an indicator function that matches 1 to rational numbers and 0 to irrationals. It is nowhere continuous.
The row and column indices of nonwhite cells are the related elements, while the different colors, other than light gray, indicate the equivalence classes (each light gray cell is its own equivalence class). In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.
Non-transitive, non-antitransitive relations include sports fixtures (playoff schedules), 'knows' and 'talks to'. The examples "is greater than", "is at least as great as", and "is equal to" are transitive relations on various sets. As are the set of real numbers or the set of natural numbers:
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 ).
The function that maps every element of A to its equivalence class is a homomorphism, and the kernel of this homomorphism is ~. The lattice Con(A) of all congruence relations on an algebra A is algebraic. John M. Howie described how semigroup theory illustrates congruence relations in universal algebra:
Holomorphic function: complex-valued function of a complex variable which is differentiable at every point in its domain. Meromorphic function: complex-valued function that is holomorphic everywhere, apart from at isolated points where there are poles. Entire function: A holomorphic function whose domain is the entire complex plane.
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, and their inverses (e.g., arcsin, log, or x 1/n).