functions which are both one-to-one and onto are also called bijective or invertible. if is bijective, then for every , there is exactly one such that (there is at least one because of surjectivity, and at most one because of injectivity). this value of is denoted ; thus is a function from to . we call the inverse of .
note that usually we propose the definition of an invertible function, propose the definition of a bijective function, then prove that a function is invertible iff it is bijective (lem-invert-1), so while bijective functions and invertible functions imply the same thing, they have different definitions.
tao's definition seems to be considering them one and the same, which i think may be a less popular approach.
another definition from college:
a function is invertible if the inverse relation is a function meaning if for every there exists a single such that meaning if for every there exists a single such that the function is called the inverse function of
because , if is invertible, then
let be a function if is invertible, then is invertible and
let be a function is invertible if and only if is total, surjective and injective.
if are invertible functions then is invertible and
the composition of invertible functions results in an invertible function
let be total functions if is injective then is injective if is surjective then is surjective if is injective and is surjective then are all invertible (what a life-saver that theorem is!)
let be total functions
- if is injective and is surjective then is injective
- if is surjective and is injective then is surjective