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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]
A function f: R → R is bijective if and only if its graph meets every horizontal and vertical line exactly once. If X is a set, then the bijective functions from X to itself, together with the operation of functional composition (∘), form a group, the symmetric group of X, which is denoted variously by S(X), S X, or X! (X factorial).
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. Compactly supported function: vanishes ...
In set theory, the Schröder–Bernstein theorem states that, if there exist injective functions f : A → B and g : B → A between the sets A and B, then there exists a bijective function h : A → B. In terms of the cardinality of the two sets, this classically implies that if | A | ≤ | B | and | B | ≤ | A |, then | A | = | B |; that is ...
In mathematics and more specifically in topology, a homeomorphism (from Greek roots meaning "similar shape", named by Henri Poincaré), [2] [3] also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function.
The emotive [note 1] function: relates to the Addresser (sender) and is best exemplified by interjections and other sound changes that do not alter the denotative meaning of an utterance but do add information about the Addresser's (speaker's) internal state, e.g. "Wow, what a view!" Whether a person is experiencing feelings of happiness ...
In English it serves a variety of functions including focusing on the object, demoting the subject and handling situations where the speaker either wants to suppress information about who the doer of the action is, or in reality does not know their identity, or when the doer is either unimportant or likely to be common knowledge.
An exclamative is a sentence type in English that typically expresses a feeling or emotion, but does not use one of the other structures. It often has the form as in the examples below of [WH + Complement + Subject + Verb], but can be minor sentences (i.e. without a verb) such as [WH + Complement] How wonderful!. In other words, exclamative ...