What is the importance of representing a (an equivalence) relation using digraphs? Is there any geometric aspect to study relations using graphs (of vertices and edges)?
2026-03-25 04:36:38.1774413398
Why study graph representations of equivalence relations?
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IF $X,Y$ are nonempty sets and $f:X\rightarrow Y$ is an onto function then $f^{-1}(y)$ defines an equivalence relation on $X$ and Can be used to define a quotient space over $X$. Topology gives you some nice tools of analyzing the space $X$ by its quotient space $Y$. But, set topology is very limited, it has no operation defined over the equivalence factors of $X$. If they are represented as vertices of a graph, the bows can represent a binaric operation over the equivalence factors. Which in turn can help define the Algebraic Structure over it.
Having an Operation and an Algebraic Structure over a space opens a window to using lots of tools that weren't available with Set Topology only.
In conclusion: When given an equivalence relation over a set, the ability to represent it as a graph gives one many more tools from Algebraic Topology/General Algebra of analyzing the space.