A bijection is a mapping that is both surjective and injective . In a bijective mapping, each element of one set corresponds to exactly one element of another set, and the inverse mapping is defined, which has the same property. Therefore, a bijective mapping is also called a one-to-one mapping (correspondence), a one-to-one mapping .
If between two sets it is possible to establish a one-to-one correspondence (bijection), then such sets are called equal-power . From the point of view of set theory , equal power sets are indistinguishable.
A one-to-one mapping of a finite set into itself is called a permutation (or permutation) of the elements of this set.
Function called a bijection (and denoted by ), If she:
- Translates various elements of a set into different elements of the set ( injectivity ). In other words,
- Any item from has its prototype ( surjectivity ). In other words,
- Identity mapping on set bijectively.
- - bijective functions from in yourself. In general, any monomial of one variable of odd degree is a bijection from in yourself.
- - bijective function of at .
- is not a bijective function, if you consider it defined on all .
- Strictly monotonous and continuous function. is a bijection of a segment on the segment .
- Function is bijective if and only if there is an inverse function such that
- If functions and bijective, then the composition of functions bijective, in this case . Briefly: the composition of bijections is a bijection. The reverse, however, is incorrect: if bijective, then we can only say that injective, and surjective.
In computer science
One-to-one connection between relational database tables based on primary keys .
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