Mapping Induces Partition on Domain

Theorem

Let $f: S \to T$ be a mapping.

Let $F$ be defined as:

$F = \set {\map {f^{-1} } x: x \in T}$

where $\map {f^{-1} } x$ is the preimage of $x$.

Then $F$ is a partition of $S$.

Proof

Let $\RR_f \subseteq S \times S$ be the relation induced by $f$:

$\tuple {s_1, s_2} \in \RR_f \iff \map f {s_1} = \map f {s_2}$

Then from Relation Induced by Mapping is Equivalence Relation, $\RR_f$ is an equivalence relation.

The result follows from Relation Partitions Set iff Equivalence.

$\blacksquare$