Relation Induced by Partition is Equivalence
Theorem
Let $\mathbb S$ be a partition of a set $S$.
Let $\RR$ be the relation induced by $\mathbb S$.
Then:
- $(1): \quad \RR$ is unique
- $(2): \quad \RR$ is an equivalence relation on $S$.
Hence $\mathbb S$ is the quotient set of $S$ by $\RR$, that is:
- $\mathbb S = S / \RR$
Proof
From the definition of Relation Induced by Partition, we define the relation $\RR$ on $S$ by:
- $\RR = \set {\tuple {x, y}: \paren {\exists T \in \mathbb S: x \in T \land y \in T} }$
Test for Equivalence
We are to show that $\RR$ is an equivalence relation.
Checking in turn each of the criteria for equivalence:
Reflexive
$\RR$ is reflexive:
\(\ds \) | \(\) | \(\ds \paren {x \in T \land x \in T}\) | Rule of Idempotence | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \tuple {x, x} \in \RR\) | Definition of $\RR$ |
$\Box$
Symmetric
$\RR$ is symmetric:
\(\ds \) | \(\) | \(\ds \tuple {x, y} \in \RR\) | ||||||||||||
\(\ds \) | \(\leadsto\) | \(\ds x \in T \land y \in T\) | Definition of $\RR$ | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds y \in T \land x \in T\) | Rule of Commutation | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \tuple {y, x} \in \RR\) | Definition of $\RR$ |
$\Box$
Transitive
$\RR$ is transitive:
\(\ds x \in T \land y \in T\) | \(\leadstoandfrom\) | \(\ds \tuple {x, y} \in \RR\) | Definition of $\RR$ | |||||||||||
\(\ds y \in T \land z \in T\) | \(\leadstoandfrom\) | \(\ds \tuple {y, z} \in \RR\) | Definition of $\RR$ | |||||||||||
\(\ds \paren {x \in T \land y \in T} \land \paren {y \in T \land z \in T}\) | \(\leadstoandfrom\) | \(\ds \tuple {x, z} \in \RR\) | Definition of $\RR$ |
$\Box$
So $\RR$ is reflexive, symmetric and transitive.
Hence, by definition, $\RR$ is an equivalence relation.
$\Box$
Test for Uniqueness
Now by definition of a partition, we have that:
- $\mathbb S$ partitions $S \implies \forall x \in S: \exists T \in \mathbb S: x \in T$
Also:
\(\ds x, y \in T\) | \(\leadsto\) | \(\ds \tuple {x, y} \in \RR\) | Definition of $\RR$ | |||||||||||
\(\ds x \in T_1, y \in T_2, T_1 \ne T_2\) | \(\leadsto\) | \(\ds \tuple{x, y} \notin \RR\) | Definition of Set Partition |
Thus $\mathbb S$ is the set of $\RR$-classes constructed above, and no other relation can be constructed in this way.
$\blacksquare$
Also see
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): $\S 7$: Relations
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.2$
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 2.4$. Equivalence classes: Example $37$
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $2$. Equivalence Relations: Exercise $4$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 10$: Equivalence Relations: Theorem $10.2$
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.3$: Theorem $5$ Part $\text {II}$
- 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 1.4$: Decomposition of a set into classes. Equivalence relations: Theorem $4$
- 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): Appendix: Elementary set and number theory
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: Equivalence Relations: $\S 17 \alpha$
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations: Theorem $6.4$
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.8$: Quotient spaces
- 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.3$: Equivalence Relations: Problem Set $\text{A}.3$: $16$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.4$: Equivalence relations
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $1$: Pairs, Relations, and Functions: Exercise $10 \ \text {(b)}$
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations
- 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.3$: Relations: Remark $2.3.1$