# Fibers

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• Jul 16th 2007, 12:29 PM
tukeywilliams
Fibers
7. $f: A \rightarrow B$ is a surjective map of sets. We want to prove that $a \sim b$ if and only if $f(a) = f(b)$ is an equivalence relation whose equivalence classes are fibers of $f$.
• Jul 16th 2007, 12:40 PM
Plato
Here it is: $
\begin{array}{l}
\left( {\forall x \in A} \right)\left[ {f(x) = f(x)} \right] \\
\left[ {f(x) = f(y)\quad \Rightarrow \quad f(y) = f(x)} \right] \\
\left[ {f(x) = f(y)\quad \& \quad f(y) = f(z)\quad \Rightarrow \quad f(x) = f(z)} \right] \\
\end{array}$