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?
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$\begingroup$How about just $$\Gamma_f = \{ (x,x):0\leq x \leq 3\} \cup \{ (x,3): 3<x\leq 8\} \cup \{ (x,x^2):x>8\}$$
$\endgroup$ $\begingroup$As a practical notation I'll recommend MPW's suggestion. But note that your proposal $$ \{ (x,y) \mid (x\in[0,3] \rightarrow y=x) \lor (x \in (3,8] \rightarrow y=3) \lor (x \in (8,\infty) \rightarrow y=x^2) \} $$ will not work -- that expression will in fact produce all of $\mathbb R\times\mathbb R$.
Namely, no matter what $x$ is, one of $x\in[0,3]$ and $x\in(3,8])$ will be false. And since an implication is always true when the antecedent is false, this means that at least one of the claims you're putting "or" between will be true no matter what $x$ and $y$ are.
If you want to write the set in this style it should be either $$ \{ (x,y) \mid (x\in[0,3] \land y=x) \lor (x \in (3,8] \land y=3) \lor (x \in (8,\infty) \land y=x^2) \} $$ or $$ \{ (x,y) \mid (x\in[0,3] \rightarrow y=x) \land (x \in (3,8] \rightarrow y=3) \land (x \in (8,\infty) \rightarrow y=x^2) \} $$
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