I am familiar with equivalence relation on a set, and its classical examples such as
forming rational/real numbers, modular arithmetic, homotopy/homology, different forms of matrices etc.
But, I wanted to know an example/application of this in Computers, where certain problem on a set is reduced to equivalence classes on the set.
In the google search, I was unable to see examples in computer science (I saw many examples in Pure Mathematics branches, which I am not considering in above problem).
For example the Myhill-Nerode theorem uses equivalence classes on strings in an alphabet to determine whether a formal language is regular or not.
A language $L$ is a set of finite strings composed of characters from some alphabet $\Sigma$. We can say that two strings $x$ and $y$ are equivalent if for any string $z$, we have $x^\frown z\in L$ if and only if $y^\frown z\in L$ (here $^\frown$ denotes string concatenation). Let's use the symbol $x\equiv_L y$ to denote that $x$ and $y$ are equivalent.
Now the Myhill-Nerode theorem states that $L$ is a regular language if and only if there are only finitely many equivalence classes. In other words, if we can find an infinite set of strings $\{x_0,x_1,x_2,\dots\}$ such that $x_i\not\equiv_L x_j$ for every $i\neq j$, then $L$ is not a regular language, and vice versa.