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- In mathematics, a preimage refers to the original set of elements in the domain of a function that gets mapped to another set of elements in the range.
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Definition: Preimage of a Set. Given a function \(f :{A}\to{B}\), and \(D\subseteq B\), the preimage \(D\) of under \(f\) is defined as \[f^{-1}(D) = \{ x\in A \mid f(x) \in D \}.\] Hence, \(f^{-1}(D)\) is the set of elements in the domain whose images are in \(C\). The symbol \(f^{-1}(D)\) is also pronounced as “\(f\) inverse of \(D\).”
If $\Map fXY$ is a function and $A\subseteq X$ and $B\subseteq Y$ are some set then the set $$\Img fA=\{f(x); x\in A\}$$ is called the image of the subset $A$ and the set $$\Pre fB=\{x; f(x)\in B\}$$ is called the preimage or inverse image of the subset $B$.
🔗. We should define these sets more precisely: 🔗. Definition 10.3.1. Image and preimage. Let f: A → B be a function, and let C ⊆ A and let . D ⊆ B. 🔗. The set f (C) = {f (x): x ∈ C} is the image of C in B. 🔗. The set f − 1 (D) = {x ∈ A: f (x) ∈ D} is the preimage of D in A or f -inverse of D. 🔗. Remark 10.3.2. Preimage of a single element.
Let f: A → B be a function, and let B0,B1 ⊆ B. Prove the following properties of preimages. Inclusions are preserved: B0 ⊆ B1 f−1(B0) ⊆f−1(B1). Unions are preserved: f−1(B0 ∪B1) = f−1(B0) ∪f−1(B1). Intersections are preserved: f−1(B0 ∩B1) = f−1(B0) ∩f−1(B1).
Apr 17, 2022 · If \(T\subseteq Y\), the preimage (or inverse image) of \(T\) under \(f\) is defined via \[f^{-1}(T):= \{x\in X \mid f(x)\in T\}.\] The image of a subset \(S\) of the domain is simply the subset of the codomain we obtain by mapping the elements of \(S\) .
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.
Oct 10, 2024 · The preimage is defined whether f has an inverse or not. Note however that if f does have an inverse, then the preimage f^(-1)(Y) is exactly the image of Y... Let f:A->B be a map between sets A and B.
