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  2. Bijection, injection and surjection - Wikipedia

    en.wikipedia.org/wiki/Bijection,_injection_and...

    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]

  3. Bijection - Wikipedia

    en.wikipedia.org/wiki/Bijection

    A function is bijective if and only if it is invertible; that is, a function : is bijective if and only if there is a function :, the inverse of f, such that each of the two ways for composing the two functions produces an identity function: (()) = for each in and (()) = for each in .

  4. Function (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Function_(mathematics)

    As a word of caution, "a one-to-one function" is one that is injective, while a "one-to-one correspondence" refers to a bijective function. Also, the statement "f maps X onto Y" differs from "f maps X into B", in that the former implies that f is surjective, while the latter makes no assertion about the nature of f. In a complicated reasoning ...

  5. List of types of functions - Wikipedia

    en.wikipedia.org/wiki/List_of_types_of_functions

    Surjective function: has a preimage for every element of the codomain, that is, the codomain equals the image. Also called a surjection or onto function. Bijective function: is both an injection and a surjection, and thus invertible. Identity function: maps any given element to itself. Constant function: has a fixed value regardless of its input.

  6. Surjective function - Wikipedia

    en.wikipedia.org/wiki/Surjective_function

    The natural logarithm function ln : (0, +∞) → R is a surjective and even bijective (mapping from the set of positive real numbers to the set of all real numbers). Its inverse, the exponential function , if defined with the set of real numbers as the domain and the codomain, is not surjective (as its range is the set of positive real numbers).

  7. Horizontal line test - Wikipedia

    en.wikipedia.org/wiki/Horizontal_line_test

    Variations of the horizontal line test can be used to determine whether a function is surjective or bijective: The function f is surjective (i.e., onto) if and only if its graph intersects any horizontal line at least once. f is bijective if and only if any horizontal line will intersect the graph exactly once.

  8. Injective function - Wikipedia

    en.wikipedia.org/wiki/Injective_function

    In mathematics, an injective function (also known as injection, or one-to-one function [1]) is a function f that maps distinct elements of its domain to distinct elements of its codomain; that is, x 1 ≠ x 2 implies f(x 1) ≠ f(x 2) (equivalently by contraposition, f(x 1) = f(x 2) implies x 1 = x 2).

  9. Function composition - Wikipedia

    en.wikipedia.org/wiki/Function_composition

    The set of all bijective functions f: X → X (called permutations) forms a group with respect to function composition. This is the symmetric group , also sometimes called the composition group . In the symmetric semigroup (of all transformations) one also finds a weaker, non-unique notion of inverse (called a pseudoinverse) because the ...