I have searched, but I haven't found an answer to this question. How are piecewise functions defined in set theory? And what are alternative ways to represent them?
Let's say we have a piecewise function $f:\mathbb{R}_+ \to \mathbb{R}$, that is defined by the equation
$$ f(x)=
\begin{cases}
x & x \in [0,3] \\
3 & x \in (3,8] \\
x^2 & x \in (8,\infty)
\end{cases}
$$
The graph of this function is the set
$$ \text{graph } f= \{ (x,y) \mid y=f(x), x \in \mathbb{R}_+ \}, $$
but how can I write out this set "graph $f$" using information from the above equation? My quess would be something like this:
$$ \{ (x,y) \mid (x\in[0,3] \rightarrow y=x) \text{ or } (x \in (3,8] \rightarrow y=3) \text{ or } (x \in (8,\infty) \rightarrow y=x^2) \} $$
Is there anything else to know about piecewise functions in set theory, other than the function's graph?
Best Answer
How about just $$\Gamma_f = \{ (x,x):0\leq x \leq 3\} \cup \{ (x,3): 3<x\leq 8\} \cup \{ (x,x^2):x>8\}$$