Definition:Induced Equivalence

Definition

Let $f: S \to T$ be a mapping.

Let $\mathcal R_f \subseteq S \times S$ be the relation defined as:

$\left({s_1, s_2}\right) \in \mathcal R_f \iff f \left({s_1}\right) = f \left({s_2}\right)$

The relation $\mathcal R_f$ is an equivalence relation.

It is known as:

• the (equivalence) relation induced by (the mapping) $f$
• the (equivalence) relation defined by (the mapping) $f$
• the (equivalence) relation associated with (the mapping) $f$.

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 8$: Functions
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 10$
• A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 1.4$: Example $1$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 6$: Example $6.6$