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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Definition 1 (Function or map). Let \(X,Y\) be non-empty sets. A rule that
assigns to each argument \(x\in
X\) a unique value \(y\in
Y\) is called a map or function from \(X\) into \(Y\). One writes for this \(y\) usually \(f(x)\).
Notation:
\[\begin{aligned}
f:X &\rightarrow Y \\
x &\mapsto f(x)
\end{aligned}\] Here, \(X\) is
called domain of \(f\), and
\(Y\) is called
codomain.
Attention 2 (Two arrows!). We use the arrow “ \(\to\) ” only between the sets, domain and codomain, and “ \(\mapsto\) ” only between the elements.
Example 3. \(f:\mathbb{N} \rightarrow \mathbb{N}\) with \(f(x)=x^2\) maps each natural number to its square.
Well-definedness
What can go wrong with the definition of a map? Sometimes, when defining a function, it is not completely clear, if this makes sense. Then one has to work and make this function well-defined.
Example: the square-root
Try to define a map \(a \to \sqrt{a}\) in a mathematically rigorous way.
Naive definition: \[\begin{aligned} \sqrt{\hphantom{x}} : \mathbb{R} &\to \mathbb{R}\\ a &\mapsto \mbox{ the solution of } x^2=a. \end{aligned}\] Problem of well-definedness: As we all know, the above equation has two (\(a>0\)), one (\(a=0\)), or zero (\(a<0\)) solutions.
First way: restrict the domain of definition and the codomain \[\mathbb{R}^+_0 = \{ a \in \mathbb{R}: a \ge 0 \}\] Then: \[\begin{aligned} \sqrt{\hphantom{x}} : \mathbb{R}^+_0 &\to \mathbb{R}^+_0\\ a &\mapsto \mbox{ the non-negative solution of } x^2=a. \end{aligned}\] This yields the classical square-root.
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