Bijektivitet

Bijektivitet is a concept in mathematics that describes a specific type of function between two sets. A function is considered bijective if it is both injective (one-to-one) and surjective (onto).

Injectivity means that every element in the codomain is mapped to by at most one element in the domain. In other words, if two elements in the domain are different, their corresponding images in the codomain must also be different.

Surjectivity means that every element in the codomain is mapped to by at least one element in the domain. This implies that the range of the function is equal to its codomain.

When a function is both injective and surjective, it is called a bijection. This means there is a perfect pairing between the elements of the domain and the elements of the codomain. For every element in the domain, there is exactly one corresponding element in the codomain, and for every element in the codomain, there is exactly one corresponding element in the domain that maps to it.

Bijective functions are important because they establish a one-to-one correspondence between sets. This allows us to count the number of elements in finite sets and compare the sizes of infinite sets. If a bijection exists between two sets, they are considered to have the same cardinality, meaning they have the same number of elements. This concept is fundamental in set theory and has applications in various areas of mathematics.