How I can define an equivalence relation?

Question

Let $f:ℝ→ℝ$ be a real analytic function. Assume that $f$ has infinitely may zeros. Let $D$ be the set values of the function $f$. I want to construct a partition of $D$ as follow: $D=D₁∪D₂$ where $D₁= \{ x∈D:f(x)=0 \} $ and $D₂= \{ x∈D:f(x)≠0 \} $. One way to do this is to define an equivalence relation. My question is: How I can define this relation?

Answer 1

A partition is the same as an equivalence relation: Given any partition $P=\{D_i:i\in I\}$ of any set $D$, you can define the equivalence relation $\sim_P$ by $$x\sim_P y \Longleftrightarrow \exists i\in I: x,y\in D_i.$$ Conversely, any equivalence relation gives you a partition of $D$ into equivalence classes.