Mappings

Mappings are a foundational concept describing how elements from one set, called the domain, are related to elements of another set, called the codomain. In mathematics, a mapping is typically a function: a rule that assigns to every input in the domain exactly one output in the codomain. When a rule is defined only for some inputs, it is a partial mapping; when it covers all inputs, it is a total mapping.

A mapping is often denoted by f: A → B, where A is the domain and B is the codomain. The image f(A) is the set of outputs that actually occur, a subset of B. For a given y in B, the preimage f^{-1}(y) is the set of inputs that map to y. The graph of a mapping is the set of ordered pairs (x, f(x)).

Mappings have several important properties. An injective (one-to-one) mapping assigns distinct outputs to distinct inputs; a surjective (onto) mapping covers the entire codomain; a bijective mapping is both injective and surjective and thus establishes a one-to-one correspondence between domain and codomain. If a mapping is bijective, it has an inverse mapping that reverses the assignment.

In computer science, a mapping often refers to a data structure that associates keys with values, such as dictionaries or maps. Operations include lookup, insertion, and deletion, with performance typically tied to the underlying implementation (for example, hash maps or balanced trees).

Mappings also appear in cartography and geographic information systems, where a cartographic mapping represents real-world features using coordinate systems and projections to translate space onto a planar surface.

Overall, mappings provide a formal way to relate elements across sets, with wide applications across mathematics, computer science, and geography.