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A mapping (or function) f from S to T, where S and T are non-empty sets, is a rule which associates, with each element of S, a unique element of T. The set S is the domain and T is the codomain of f. The phrase ‘f from S to T’ is written ‘f: ST’. For s in S, the unique element of T that f associates with s is the image of s under f and is denoted by f(s). If f(s)=t, it is said that f maps s to t, written f: st. The subset of T consisting of those elements that are images of elements of S under f, that is, the subset { t | t=f(s), for some s in S}, is the image (or range) of f, denoted by f(S). For the mapping f: ST, the subset {(s, f(s) | sS} of the Cartesian product S×T is the graph of f. The graph of a mapping has the property that, for each s in S, there is a unique element (s, t) in the graph. Some authors define a mapping from S to T to be a subset of S×T with this property; then the image of s under this mapping is defined to be the unique element t such that (s, t) is in this subset.

Subjects: Mathematics.

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