irányhalmaz

Irányhalmaz, in Hungarian mathematics, refers to the set of possible outputs produced by a function or mapping. It is commonly understood as the image or range of a function, i.e., the subset of the codomain consisting of all values that can be obtained from some input.

Formally, if f is a function from X to Y, denoted f: X → Y, then the irányhalmaz of f is Im(f) = { f(x) : x ∈ X }. This irányhalmaz is a subset of Y. If Im(f) = Y, the function is surjective (onto). If X and Y have additional structure, such as topology or vector space structure, the irányhalmaz inherits or interacts with that structure accordingly.

Examples illustrate the concept. For f: R → R with f(x) = x^2, the irányhalmaz (image) is [0, ∞). For a linear map f: R^2 → R^2 given by f(x, y) = (x + y, x − y), the irányhalmaz is all of R^2, since every point in the plane can be written as f(x, y) for some (x, y).

In broader usage, the term can also appear in contexts describing the set of attainable directions or outputs in geometric or optimization problems, where the focus is on the possible results from varying inputs within a given domain. See also: domain, codomain, image, preimage, surjectivity.