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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]
Functions which satisfy property (4) are said to be "one-to-one functions" and are called injections (or injective functions). [2] With this terminology, a bijection is a function which is both a surjection and an injection, or using other words, a bijection is a function which is both "one-to-one" and "onto". [3]
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.
That is, a "topic outline" is an outline composed of topics only, so as soon as an editor comes along and adds an annotation or sentence or descriptive lead or section lead, it ceases to be a topic outline and then has an erroneous title.
If we assume the axiom of choice, then a pair of surjective functions and also implies the existence of a bijection. We construct an injective function h : B → A from f − 1 {\displaystyle f^{-1}} by picking a single element from the inverse image of each point in B {\displaystyle B} .
Any involution is a bijection.. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (x ↦ −x), reciprocation (x ↦ 1/x), and complex conjugation (z ↦ z) in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the ...
An outline, also called a hierarchical outline, is a list arranged to show hierarchical relationships and is a type of tree structure. An outline is used [1] to present the main points (in sentences) or topics of a given subject. Each item in an outline may be divided into additional sub-items.
Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space.