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In mathematics, for a function :, the image of an input value is the single output value produced by when passed . The preimage of an output value y {\displaystyle y} is the set of input values that produce y {\displaystyle y} .
A rectangular grid (top) and its image under a conformal map f (bottom). It is seen that f maps pairs of lines intersecting at 90° to pairs of curves still intersecting at 90°. A conformal map is a function which preserves angles locally. In the most common case the function has a domain and range in the complex plane. More formally, a map,
Mathematical visualization is used throughout mathematics, particularly in the fields of geometry and analysis. Notable examples include plane curves, space curves, polyhedra, ordinary differential equations, partial differential equations (particularly numerical solutions, as in fluid dynamics or minimal surfaces such as soap films), conformal ...
A function : is monotone in this topological sense if and only if it is non-increasing or non-decreasing, which is the usual meaning of "monotone function" in real analysis. A function between topological spaces is (sometimes) called a proper map if every fiber is a compact subspace of its domain. However, many authors use other non-equivalent ...
In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable.
Semialgebraic geometry is the study of semialgebraic sets, i.e. real-number solutions to algebraic inequalities with-real number coefficients, and mappings between them. The most natural mappings between semialgebraic sets are semialgebraic mappings , i.e., mappings whose graphs are semialgebraic sets.
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(L1) For every proper ideal I and a prime ideal P, there exists an x in R - P such that (I : x) is the pre-image of I R P under the localization map R → R P. (L2) For every ideal I, the set of all pre-images of I S −1 R under the localization map R → S −1 R, S running over all multiplicatively closed subsets of R, is finite.