Biyection

Biyection is not a standard term in formal mathematics. In most contexts, it is a misspelling or informal coinage of the term bijection. When encountered, it is generally treated as referring to the same concept or as a typographical error.

A bijection between sets X and Y is a function f: X -> Y that is both injective and surjective. An injective function maps distinct elements of X to distinct elements of Y, meaning no two different inputs share the same output. A surjective function covers every element of Y, meaning for every y in Y there exists some x in X with f(x) = y. When both conditions hold, the function creates a perfect one-to-one correspondence between X and Y.

Because a bijection has a one-to-one correspondence, it is invertible. If f: X -> Y is a bijection, there exists a unique inverse function f^{-1}: Y -> X such that f^{-1}(f(x)) = x for all x in X and f(f^{-1}(y)) = y for all y in Y. Consequently, a bijection implies that X and Y have the same cardinality.

Examples illustrate the concept. The map f: {1, 2, 3} -> {a, b, c} defined by f(1) = a, f(2) = b, f(3) = c is a bijection. The function g: Z -> Z given by g(n) = n + 1 is also a bijection, with inverse g^{-1}(m) = m - 1.

Importance and uses of bijections include establishing equal cardinalities of sets, constructing inverses and isomorphisms, and describing permutations and coordinate correspondences in various mathematical structures.

If you encounter "biyection" in a source, treat it as a probable typo or nonstandard usage of bijection unless the author explicitly defines it differently. See also bijection, injection, surjection, isomorphism.