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A relation that is functional and total. For example, the red and green relations in the diagram are functions, but the blue and black ones are not. An injection [d] A function that is injective. For example, the green relation in the diagram is an injection, but the red, blue and black ones are not. A surjection [d] A function that is surjective.
Functional programming is the programming paradigm consisting of building programs by using only subroutines that behave like mathematical functions. For example, if_then_else is a function that takes three functions as arguments, and, depending on the result of the first function (true or false), returns the result of either the second or the ...
Nowhere continuous function: is not continuous at any point of its domain; for example, the Dirichlet function. Homeomorphism: is a bijective function that is also continuous, and whose inverse is continuous. Open function: maps open sets to open sets. Closed function: maps closed sets to closed sets.
A function is bijective if it is both injective and surjective. A bijective function is also called a bijection or a one-to-one correspondence (not to be confused with one-to-one function, which refers to injection). A function is bijective if and only if every possible image is mapped to by exactly one argument. [1]
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
Mathematical relations fall into various types according to their specific properties, often as expressed in the axioms or definitions that they satisfy. Many of these types of relations are listed below.
Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function. Kronecker delta function: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.
The equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called Con X by convention. The canonical map ker : X^X → Con X, relates the monoid X^X of all functions on X and Con X. ker is surjective but not injective. Less formally, the equivalence relation ker on X, takes each function f : X → X to its ...