Equivalence relation induced by a function

Definition

Let $f \colon A \to B$ be a function. The equivalence relation induced by $f$ is the relation $\sim_f$ on $A$ with $$a \sim_f a' \iff f(a) = f(a')$$ for all $a, a' \in A$.

Remarks

Another name is the kernel of $f$. The relation $\sim_f$ is an equivalence relation, because equality is one. Its classes are the nonempty preimages $f^{-1}(\{b\})$ of the points $b \in B$. The function $f$ is injective exactly when $\sim_f$ is equality.

Explanations

L1 human

Take the function that sends each person to their birthday. Two people are equivalent under it when they have the same birthday. The function forgets everything except its value, and the relation records what it forgets.