Image and Preimage
Via images and preimages we describe how functions work on sets.
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Study Image and Preimage
For every well-defined map \(f: X\to Y\) and \(A\subset X\), \(B \subset Y\) we are interested in the following sets:
Definition 1. Let \(f: X\rightarrow Y\) be a function and \(A\subset X\) and \(B\subset Y\) some sets.
\[f(A):= \{ f(x): x\in A\}\] is called the image of \(A\) under \(f\).
\[f^{-1}(B):= \{ x\in X: f(x) \in B
\}\] is called the preimage of \(B\) under \(f\). 
Note that the preimage can also be the empty set if none of the elements in \(B\) are “hit” by the map.
To describe the behaviour of a map, the following sets are very important:
Definition 2 (Range and fiber). Let \(f: X\rightarrow Y\) be a map. Then \[\begin{aligned} \mathrm{Ran}(f) &:= f(X) = \{ f(x) : x \in X \} \end{aligned}\] is called the range of \(f\). For each \(y\in Y\) the set \[\begin{aligned} f^{-1}(\{y \}) &:= \{ x \in X : f(x) = y \} \end{aligned}\] is called a fiber of \(f\).
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