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Definition: Image of a Set; Definition: Preimage of a Set; Summary and Review; Exercises ; One-to-one functions focus on the elements in the domain. We do not want any two of them sharing a common image. Onto functions focus on the codomain. We want to know if it contains elements not associated with any element in the domain.
Here's interesting application of inverse image: Given two functions: $ f : R \times R \rightarrow R, g : R \rightarrow 2, g=([n_0..n_1] \mapsto 1)$, the composition of them gives characteristic function $ h : R \times R \rightarrow 2$.
The preimage of an output value is the set of input values that produce . More generally, evaluating f {\displaystyle f} at each element of a given subset A {\displaystyle A} of its domain X {\displaystyle X} produces a set, called the " image of A {\displaystyle A} under (or through) f {\displaystyle f} ".
Oct 10, 2024 · Let f:A->B be a map between sets A and B. Let Y subset= B. Then the preimage of Y under f is denoted by f^(-1)(Y), and is the set of all elements of A that map to elements in Y under f. Thus f^(-1)(Y)={a in A|f(a) in Y}. (1) One is not to be mislead by the notation into thinking of the preimage as having to do with an inverse of f. The preimage is defined whether f has an inverse or not. Note ...
10.3. Images and preimages of sets. 🔗. When we defined the function , f: A → B, we said that if f (a) = b then we called b the image of a under . f. This idea can be extended quite naturally to think of the image of a set of points. Also, given an element , b ∈ B, we can ask for all the elements of A that map to it.
The preimage of an element y in set B is the set of all elements x in set A such that f(x) = y. The preimage is denoted as f^(-1)(y), where the (-1) superscript does not denote the inverse function but represents the preimage operation. To visualize this, let’s consider a simple function f(x) = x^2, which maps elements from set A (real ...
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Apr 17, 2022 · Notice that we defined the preimage of a subset of the codomain regardless of whether \(f^{-1}\) is a function or not. In particular, for \(T\subseteq Y\), \(f^{-1}(T)\) is the set of elements in the domain that map to elements in \(T\). As a special case, \(f^{-1}(\{y\})\) is the set of elements in the domain that map to \(y\in Y\).
