Injektiivsust

Injektiivsust, often translated as injectivity, is a fundamental concept in mathematics, particularly within abstract algebra and set theory. A function f from a set A to a set B is called injective if distinct elements in A are mapped to distinct elements in B. In simpler terms, if two elements in the domain A are different, then their corresponding images in the codomain B must also be different. This property can be formally stated as: for any x1, x2 in A, if f(x1) = f(x2), then x1 = x2. Conversely, if x1 is not equal to x2, then f(x1) is not equal to f(x2).

Injective functions are also known as one-to-one functions. This means that each element in the range of the function is the image of at most one element in the domain. Injective functions are crucial for defining concepts like isomorphisms in abstract algebra, where they preserve the structure of algebraic objects. In set theory, an injective function establishes a way to "count" or compare the sizes of sets, particularly when dealing with infinite sets. If there exists an injective function from set A to set B, it implies that the cardinality of A is less than or equal to the cardinality of B. The existence of a bijection (a function that is both injective and surjective) between two sets signifies that they have the same cardinality.