injektywna

Injektywna is the feminine form of the mathematical adjective used in some languages to describe a function that is injective, i.e., one-to-one. A function f: X → Y is injective if different inputs map to different outputs, meaning that x1 ≠ x2 implies f(x1) ≠ f(x2). Equivalently, f(x1) = f(x2) implies x1 = x2.

Several equivalent characterizations exist. For every x1, x2 in X, f(x1) = f(x2) only when x1 = x2. Another way to express this is that the preimage of any element in the image f(X) contains at most one element. A common theoretical statement is that f has a left inverse on its image: there exists a function g: Y → X such that g(f(x)) = x for all x in X.

Examples help illustrate the concept. The function f: Z → Z defined by f(n) = 2n is injective, since no two integers map to the same value. The function f: Z → Z given by f(n) = n^2 is not injective, because f(1) = f(-1) = 1. On the real numbers, f(x) = e^x is injective, while f(x) = x^2 is not injective.

In finite settings, injectivity implies a cardinality constraint: if f: X → Y is injective, then |X| ≤ |Y|. If the sets have the same finite size and f is injective, then f is also surjective and hence bijective. Injectivity is a fundamental property in many areas of mathematics, with implications for inverses, compositions, and structure-preserving mappings.