Via images and preimages we describe how functions work on sets.

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Concept |
Content |

Sets |
Sets are the basic building blocks for a lot of mathematics. In order to rigorously define numbers and doing real analysis, we need to know how to work with sets. |

Maps |
Maps are the mathematical formulation of a machine that gets inputs and generate outputs. On both sides, sets are needed for the domain and the codomain. |

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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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