3: Mappings
Published online by Cambridge University Press: 05 June 2012
Summary
A mapping (or function) f from a set X to a set Y is a relation between X and Y (i.e. a subset of X × Y) with the properties
(a) given x∈X there is some y ∈ Y with (x, y) ∈ f,
(b) if(x, y1), (x, y2)∈ f then y1 = y2.
We denote such a mapping f by writing f:X → Y, and if (x, y)∈ f we use the notation y = f(x). The set X is called the domain of f; and the subset {y ∈ Y| (∃x ∈ X)y = f(x)} of Y is called the image of f and is denoted by Im f. When it is clear what the sets X and Y are, we often write the mapping f in the form x → f(x). If A ⊆ X then f(A) is defined to be {y ∈ Y | (∃x∈A)y = f(x)}. In particular, f(X) = Im f. Two mappings f:A → B and g : C → D are said to be equal if A = C, B = D and (∀x ∈ A)f(x) = g(x).
Given f : X → Y and g : Y → Z, the composite g ∘ f is the mapping g ∘ f: X → Z defined by (g ∘ f)(x) = g[f(x)] for every x ∈ X. For mappings f, g and h we recall that the associative law (f ∘ g)∘ h = f ∘ (g ∘ h) holds whenever these multiple composites are defined. Also, when f: X → X we use the usual notation f2 for f ∘ f, and in general fn for f ∘ f … ∘ f(n terms).
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- Algebra Through PracticeA Collection of Problems in Algebra with Solutions, pp. 13 - 23Publisher: Cambridge University PressPrint publication year: 1984