Definition:Equivalence Class/Also known as

Equivalence Class: Also known as

The equivalence class of $x$ under $\RR$ can also be referred to as:

the equivalence class of $x$ determined by $\RR$

the equivalence class of $x$ with respect to $\RR$

the equivalence class of $x$ modulo $\RR$.

It can be stated more tersely as the $\RR$-equivalence class of $x$, or just the $\RR$-class of $x$.

The term equivalence set can also occasionally be found for equivalence class.

Some sources, for example P.M. Cohn: Algebra Volume 1 (2nd ed.), use the term equivalence block.

Sources

• 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Relations
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: Equivalence Relations: $\S 17$
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations
• 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.3$: Equivalence Relations
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 17$: Equivalence classes
• 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