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Bijective composition: the first function need not be surjective and the second function need not be injective. 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
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.
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.
Another way of defining the same notion is to say that a partial bijection from A to B is any relation R (which turns out to be a partial function) with the property that R is the graph of a bijection f:A′→B′, where A′ is a subset of A and B′ is a subset of B. [5]
This is how I have memorised these words: if a function f:X->Y is injective, then the image of the domain X is a subset in the codomain Y but not necessarily equal to the whole codomain (or, more precisely, a function f:X->Y is injective iff the function f defines a bijection between the set X and a subset in Y); as the word "sur" means "on" in ...
Any surjective function induces a bijection defined on a quotient of its domain by collapsing all arguments mapping to a given fixed image. More precisely, every surjection f : A → B can be factored as a projection followed by a bijection as follows.
A faithful functor need not be injective on objects or morphisms. That is, two objects X and X′ may map to the same object in D (which is why the range of a full and faithful functor is not necessarily isomorphic to C), and two morphisms f : X → Y and f′ : X′ → Y′ (with different domains/codomains) may map to the same morphism in D.
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 ...