Image (mathematics)

In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. The image of a subset A ⊆ X under f is the subset of Y defined by

Related Topics:
Mathematics - Set - Function - Subset

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:f = {y ∈ Y | y = f(x) for some x ∈ A}

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Notice that the range of f is the image f of its domain X.

Related Topics:
Range - Domain

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An alternative notation for f, favored by set theorists, is f "A.

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When there is no risk of confusion, f is sometimes denoted simply f(A).

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With this definition, the image f becomes a function whose domain is the set of all subsets of X (also known as the power set of X) and whose codomain is the power set of Y. Note that the same notation is used for the original function f and its image. This is a common convention; the intended usage must be inferred from context.

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The preimage or inverse image of a set B ⊆ Y under f is the subset of X defined by

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:f −1 = {x ∈ X | f(x) ∈ B}

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The inverse image of a singleton, f −1, is called a fiber or fibre, or level set.

Related Topics:
Singleton - Level set

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Again, in a context where there is no risk of confusion, we may denote f −1 by f −1(B), and think of f −1 as a function from the power set of Y to the power set of X.

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Note that f −1 should not be confused with the inverse function. The two only coincide if f is bijective.

Related Topics:
Inverse function - Bijective

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Looking at it the other way, f can be seen as a family of sets indexed by Y. An obvious extension of this idea is that of a fibred category.

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