Equivalence relation induced by a function

Every function $f:X\to Y$ induces an equivalence relation on $X$ by $x\sim y$ exactly when $f(x)=f(y)$. Reflexivity, symmetry, and transitivity follow from the corresponding properties of equality in $Y$; the equivalence classes are the nonempty fibers of $f$.

Positive integers are equivalent when they have the same finite set of prime factors. Equivalently, each divides a positive power of the other. This equivalence relation is induced by the radical of an integer; its equivalence classes correspond to finite prime supports. The empty support gives the singleton $\{1\}$, and every nonempty support gives an infinite class by varying exponents.