# Definition:Equivalence Relation Induced by Mapping

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## Contents

## Definition

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

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

- $\tuple {s_1, s_2} \in \RR_f \iff \map f {s_1} = \map f {s_2}$

Then $\RR_f$ is known as the **equivalence (relation) induced by $f$**.

## Also known as

The **equivalence induced by $f$** is variously known as:

- the
**(equivalence) relation (on $S$) induced by (the mapping) $f$** - the
**(equivalence) relation (on $S$) defined by (the mapping) $f$** - the
**(equivalence) relation (on $S$) associated with (the mapping) $f$** - the
**equivalence kernel of $f$**.

## Also see

- Relation Induced by Mapping is Equivalence Relation for a demonstration that $\RR_f$ is indeed an equivalence relation.

## Sources

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts*... (previous) ... (next): Introduction $\S 3$: Equivalence relations - 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 8$: Functions - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 10$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Factoring Functions: Theorem $10$ - 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $\S 1.4$: Decomposition of a set into classes. Equivalence relations: Example $1$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 3$: Equivalence relations and quotient sets: Quotient sets - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations: Example $6.6$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.4$: Equivalence relations - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations: Exercise $3.4$