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Any function $f$ with domain $X$ induces an equivalence relation on $X$, with classes $$\{f^{-1}(\{y\})\,:\, y \in \operatorname{im}f\;\} .$$

Is there a name for this equivalence?


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Well, the graph of the equivalence relation is the kernel pair of $f$. I guess you could call it the ‘pullback of the equality relation from the codomain’ if you want... – Zhen Lin Dec 8 '11 at 12:29
I do not know about the name, it also seems that the notation varies quite a lot, e.g $X/f$ in Example 2.2.2 here, i think I've seen $\theta_f$ too. Perhaps it is good to know, that every equivalence relation can be obtained in this way. This fact can be found in many standard textbook, to mention at least one - the first one google told me about - I'll give a link to Schechter's Classical and nonclassical logics p.100. – Martin Sleziak Dec 9 '11 at 12:17
up vote 7 down vote accepted

If $f:X\to Y$ then $f^{-1}(y)$ is called a fiber. The equivalence relation you defined is the partition of fibers of the function $f$.

I have not seen an explicit name for this, however it is common to say that $x_1$ and $x_1$ are in the same fiber, or explicitly in the fiber of $y$.

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"Having the same value under $f$ ". For instance if $f$ computes the absolute value, the equivalence is having the same absolute value. In some cases there is a special name, like "congruence modulo $n$" instead of "having the same remainder after division by $n$", but I don't think there is any dedicated terminology for the general case.

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As you point out, any function $f$ induces an equivalence relation $E_f$ on its domain and this relation, according to Algebra, 3rd Ed by Birkhoff & Mac Lane, is called the equivalence kernel of $f$. This term, together with the notation $E_f$ supplied above, is defined on page 33.

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Thanks, that's a very easy-to-remember notation/terminology. – kjo Feb 16 '12 at 13:51

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