Search results
Results From The WOW.Com Content Network
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 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 .
In combinatorics, bijective proof is a proof technique for proving that two sets have equally many elements, or that the sets in two combinatorial classes have equal size, by finding a bijective function that maps one set one-to-one onto the other. This technique can be useful as a way of finding a formula for the number of elements of certain ...
Grothendieck's proof of the theorem [3] [4] is based on proving the analogous theorem for finite fields and their algebraic closures.That is, for any field F that is itself finite or that is the closure of a finite field, if a polynomial P from F n to itself is injective then it is bijective.
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).
For functions of a single variable, the theorem states that if is a continuously differentiable function with nonzero derivative at the point ; then is injective (or bijective onto the image) in a neighborhood of , the inverse is continuously differentiable near = (), and the derivative of the inverse function at is the reciprocal of the derivative of at : ′ = ′ = ′ (()).
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 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 ...