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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]
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.
A function is bijective if and only if it is both injective (or one-to-one)—meaning that each element in the codomain is mapped from at most one element of the domain—and surjective (or onto)—meaning that each element of the codomain is mapped from at least one element of the domain.
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 ...
In particular, the identity function is always injective (and in fact bijective). If the domain of a function is the empty set , then the function is the empty function , which is injective. If the domain of a function has one element (that is, it is a singleton set ), then the function is always injective.
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 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).
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 ...