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Injective composition: the second function need not be injective. A function is injective (one-to-one) if each possible element of the codomain is mapped to by at most one argument. Equivalently, a function is injective if it maps distinct arguments to distinct images. An injective function is an injection. [1] The formal definition is the ...
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]
Injective function: has a distinct value for each distinct input. Also called an injection or, sometimes, one-to-one function. Also called an injection or, sometimes, one-to-one function. In other words, every element of the function's codomain is the image of at most one element of its domain.
In other words, the function value f(x) in the codomain X is always the same as the input element x in the domain X.The identity function on X is clearly an injective function as well as a surjective function (its codomain is also its range), so it is bijective.
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.
The composition of one-to-one (injective) functions is always one-to-one. Similarly, the composition of onto (surjective) functions is always onto. It follows that the composition of two bijections is also a bijection. The inverse function of a composition (assumed invertible) has the property that (f ∘ g) −1 = g −1 ∘ f −1.
This example is injective in each input separately, because the functions f x and f y are always injective. However, it's not injective in both variables simultaneously, because (for example) f (2,4) = f (1,2). One can also consider partial binary functions, which may be defined only for certain values of the inputs.
Mathematically, a bidirectional map can be defined a bijection: between two different sets of keys and of equal cardinality, thus constituting an injective and surjective function: { ∀ x , x ′ ∈ X , f ( x ) = f ( x ′ ) ⇒ x = x ′ ∀ y ∈ Y , ∃ x ∈ X : y = f ( x ) ⇒ ∃ f − 1 ( x ) {\displaystyle {\begin{cases}&\forall x,x ...