Biyections

Biyections is not a standard term in mathematics and is usually a misspelling or alternate spelling of bijections. In standard usage, a bijection is a function that is both injective and surjective, providing a one-to-one correspondence between two sets.

Formally, a function f: A → B is a bijection if every element of B is the image of exactly one element of A. Equivalently, f is injective (distinct elements of A map to distinct elements of B) and surjective (every element of B is the image of some element of A). If a bijection exists, it has an inverse function f^{-1}: B → A, satisfying f^{-1}(f(a)) = a and f(f^{-1}(b)) = b for all a ∈ A, b ∈ B.

Key properties include that bijections preserve cardinality: A and B must have the same size, finite or infinite. The inverse of a bijection is itself a bijection, and the composition of bijections is a bijection. Bijections are central to transferring structure between sets, such as reindexing elements or establishing equivalence of counting problems.

Examples:

- The identity map id: N → N, id(n) = n, is a bijection.

- f: N → the even naturals 2N given by f(n) = 2n is a bijection.

- g: {a, b, c} → {1, 2, 3} with g(a) = 1, g(b) = 2, g(c) = 3 is a bijection.

Non-examples include maps that fail injectivity or surjectivity, such as h(n) = n^2: N → N is not surjective, hence not a bijection.

History and usage note that bijection is the standard term; biyection is rarely accepted and typically results from typographical error or informal usage.