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I have a finite set $A$ of real numbers. Then I have a function $F: \mathbb{R} \rightarrow \mathbb{Z}$ that maps real numbers to integers.

I would like to split A into subsets such that for all elements in a subset, the function $F$ maps to the same integer. (We would have max. $|A|$ such subsets, if all $x \in A$ map to different integers, min. 1 such subsets if they map all to the same integer)

I would like to say something like "All subsets of A, such that for every subset the elements in that subsets map to the same integer regarding function $F$", but I have no idea how to write this down nicely. Any ideas?

user3207838
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3 Answers3

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Partition $A$ into subsets $A_k$ with $F(x)=k$ for all $x\in A_k$.

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"We consider those sets $X \subseteq A$ such that $\exists k \in \mathbb{Z}.\forall x \in X. F(x) = k$."

Hans Hüttel
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The technical term is fiber. You can say "Consider the fibers of $f$ in $A$".

A fiber of $f: X \to Y$ is just a set $f^{-1}(y) = \{ x \in X : f(x)=y \}$ for $y \in Y$.

It is always true that $X = \bigcup_{y \in Y} X_y$ where $X_y=f^{-1}(y)$. This is a partition of $X$ into disjoint parts.

In your case, by considering $f|A$, you get $A = \bigcup_{k \in \mathbb Z} A_k$, where some (most) $A_k$ will be empty.

lhf
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