mapsTo

Mapsto is a mathematical notation used to indicate a mapping from elements of one set to elements of another, typically written to express how a specific input is sent to a specific output. It is often denoted by the symbol ↦ or described in text as “maps to.” In context, one may see an expression like x maps to f(x), or x ↦ f(x), to convey that the input x is assigned the value f(x) by a rule or function.

In formal language, a function f from X to Y is a set of ordered pairs (x, f(x)) with x in X and f(x) in Y. The mapsto notation provides a compact way to describe the rule without listing all pairs. In textual form, it is common to say “x maps to f(x)” or “x ↦ f(x).” In LaTeX, the mapsto symbol is produced by \mapsto, yielding x \mapsto f(x).

Mapsto is distinct from the arrow used to denote the type of a function, X → Y, which indicates a placeholder for outputs given inputs. The mapsto arrow emphasizes the rule’s action on individual elements rather than the overall mapping from domain to codomain. It is widely used in definitions, examples, and set-builder form, such as { x ↦ f(x) | x ∈ X }, to describe a function rule succinctly.

In broader contexts, mapsto appears in areas like algebra, analysis, and category theory to convey explicit functional rules and constructions, underscoring the idea that every x in the domain is associated with a unique y in the codomain.