properties of bijections
. The identity function is the bijection from to .
If , then . If is a bijection, then its inverse function is also a bijection.
If , , then . If and are bijections, so is the composition .
If , , and , then .
If , , then . If and are bijections, so is , given by .
. The function given by is a bijection.
If and , then .
Suppose and are bijections. Define as follows: for any function , let . is a well-defined function. It is one-to-one because and are bijections (hence are cancellable). For any , it is easy to see that , so that is onto. Therefore is a bijection from to . ∎
, where is the powerset of , and is the set of all functions from to .
For every , define by
Then , defined by is a well-defined function. It is one-to-one: if for , then iff , so . It is onto: suppose , then by setting , we see that . As a result, is a bijection. ∎
Remark. As a result of property 9, we sometimes denote the powerset of .
|Title||properties of bijections|
|Date of creation||2013-03-22 18:50:41|
|Last modified on||2013-03-22 18:50:41|
|Last modified by||CWoo (3771)|