Mapping

A mapping is an assignment from one set to another. In most undergraduate contexts, “mapping” and Function mean the same thing: each input in the Domain is assigned exactly one output in a codomain.

Notation:

\qquad x\mapsto f(x).$$ Here $X$ is the domain and $Y$ is the codomain. The actual outputs reached form the image/range, which may be smaller than $Y$; see [[Codomain and range]]. ## Why use the word mapping? “Function” often suggests a numerical formula such as $f(x)=x^2$. “Mapping” is more general and geometric: it can send vectors to vectors, shapes to shapes, states to states, or fields to fields. A rotation of the plane, a change of coordinates, and a time-evolution rule are all mappings. ## Mapping properties - [[Injective function|Injective]]: distinct inputs go to distinct outputs. - [[Surjective function|Surjective]]: every element of the codomain is hit. - [[Bijective function|Bijective]]: both, so the mapping can be reversed by an [[Inverse functions|inverse function]]. ## Physics intuition A dynamics law can be viewed as a mapping from an initial state to a later state. A coordinate transform maps one description of the same physical situation to another. The maths only works cleanly when the source set, target set, and restrictions are explicit. Related: [[Relation]], [[Set]], [[Function]].