niebijekcj

Niebijekcja, sometimes interpreted as niebijekcja or described as niebijekcja, is a term used in Polish mathematical language to denote a function that is not a bijection. A bijection is a function that is both injective (one-to-one) and surjective (onto), establishing a perfect correspondence between its domain and codomain. Therefore, a niebijekcja refers to any function that fails to meet one or both of these criteria. The term highlights the absence of a one-to-one correspondence rather than specifying which property is violated.

The concept encompasses several scenarios. A function may be not injective, meaning two or more inputs share the same output. It may also be not surjective, meaning some elements of the codomain are never achieved as outputs. It can also fail both properties simultaneously. Distinguishing between these failures helps in understanding the structure of mappings and in proving the existence or nonexistence of inverses.

Common examples illustrate the idea. The function f: R -> R given by f(x) = x^2 is not a bijection because it is not injective (f(-1) = f(1)) and its image is [0, ∞), not all of R. The function g: {1,2} -> {1,2}, with g(1) = 1 and g(2) = 1, is not injective and not surjective (2 is not hit). Conversely, the function h: N -> N, h(n) = n + 1, is injective but not surjective, hence also a niebijekcja.

In mathematical arguments, identifying a niebijekcja helps determine when a bijection cannot exist between sets or structures and guides reasoning about cardinalities and function properties. See also bijection, injection, and surjection. Note: the standard Polish term for this concept is niebijekcja; "niebijekcj" appears to be a truncated form.