Definition:Equivalence Class/Notation

Definition

The notation used to denote an equivalence class varies throughout the literature, but is often some variant on the square bracket motif $\eqclass x \RR$.

The symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ is a modified version of an attempt to reproduce the heavily-bolded $\sqbrk x_\RR$ found in 1967: George McCarty: Topology: An Introduction with Application to Topological Groups.

Other variants, with selected examples of texts which use those variants:

• 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts uses $\overline x$ for $\eqclass x \RR$.

• 1965: J.A. Green: Sets and Groups uses $E_x$ for $\eqclass x \RR$.

• 1965: Seth Warner: Modern Algebra uses $\bigsqcup_\RR \mkern {-28 mu} {\raise 1pt x} \ \ $ for $\eqclass x \RR$, which is even more challenging to render in our installed version of $\LaTeX$ than $\eqclass x \RR$ itself.

• 1971: Allan Clark: Elements of Abstract Algebra uses $\boldsymbol [ x \boldsymbol ]_\RR$.

• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis uses $\map {p_\RR} x$.

• 1975: T.S. Blyth: Set Theory and Abstract Algebra uses $x / \RR$ for $\eqclass x \RR$ (compare the notation for quotient set). This source also suggests $x_\RR$ as a variant.

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) uses $\map \pi x$ for $\eqclass x \RR$.

Sources

• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 6$. Indexed families; partitions; 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