If $R\subseteq A\times A$ is an equivalence relation (i.e., a relation that is reflexive on $A$, symmetric, and transitive), then for each element $x\in A$, the subset $[x]_R=\{y\in A: \langle x,y\rangle\in R\}$ of $A$ is called an equivalence class of $A$. In set theory, the word class usually refers to a collection of elements which is not necessarily a set. I wonder why $[x]_R$ was not called an equivalence set or an equivalence subset.

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    $\begingroup$ FYI, I suspect bringing up the current mostly standard distinction between "set" and "class" is a bit ahistorical, as probably few mathematicians before the 1940s would have been aware of such things or cared. Maybe not even until the mid 1960s, when Cohen's set theory work and the emergence of category theory made this distinction more mainstream. Prior to this, "class" was likely used like "collection" is, namely as an alternative word for set when one wants to distinguish (for example) an assemblage of group elements from an assemblage of subgroups. (continued) $\endgroup$ – Dave L Renfro Mar 28 '20 at 7:52
  • $\begingroup$ Also, the phrase "class equation" in group theory goes back to the early 20th century (maybe into the 19th century; I haven't investigated this), and thus the use of "class" may simply be derived from that usage. $\endgroup$ – Dave L Renfro Mar 28 '20 at 7:53
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    $\begingroup$ @DaveLRenfro Von Neumann used "equivalence class" in some algebraic contexts in late 1920-s, and he introduced the set/class distinction explicitly in 1925. Weber introduced "class fields" in 1891. So I think your second guess is more likely, while the transition to sets occurred in the foundational contexts, where it mattered, in concrete areas preserving pre-existing usage was preferred. $\endgroup$ – Conifold Mar 28 '20 at 19:47

One detailed account of the history of equivalence relations and associated terminology is Equivalence: An Attempt at a History of the Idea by Ashgari largely based on Fowler's posts on the Historia Mathematica forum. The terminology was a long time in the making, with "equivalence relation" appearing much earlier than "equivalence class". Although neither the sources nor Ashgari directly answer why the choice of "class" was preferred there is one clue as to the possible reason.

Even before "equivalence relation" was so named abstractly by Jourdain, a particular use of "equivalence" and its classes was prominent, and remained so for a while after. It is the equivalence of sets by equipotence or cardinality, the relation being existence of a bijective correspondence between them, introduced by Hume and popularized by Cantor. In this case, the equivalence classes (of all sets with the same cardinality) are not themselves sets. As this seemed to be the paradigmatic example, by the time abstract terminology was being established the authors might have favored a combination of words they were already used to hearing. And this way there was no need to explain that in the case of cardinality the "equivalence sets" are not sets.

Von Neumann introduced the set/class distinction in set theory explicitly in 1925 (it became the basis of a set theory axiomatization NBG, alternative to classless ZFC), but in 1926 and 1929 he used the expression "equivalence class" in some algebraic contexts nonetheless. Aside from cardinality, there was some pre-existing terminology with "classes" in algebra, e.g. Weber introduced "class field" in 1891, see Miller's Earliest Known Uses, Jordan in a paper on group characters talked of "classes of conjugate substitutions" (to become conjugacy classes) in 1907. So it seems that the terminological transition from classes to sets only happened narrowly, where it mattered, and older terminology was generally preferred where it was already established.

Here are some relevant passages from Ashgari:

"I call a relation which is reflexive, symmetrical, and transitive an isoid relation. Philip Jourdain (1912) presented his article On isoid relations and theories of irrational number to the Fifth International Congress of Mathematics, starting with the sentence above. At the time, the process of “definition by abstraction” (Russell 1903, pp. 219-220) was quite well established but the term “equivalence” was mainly attached to the context of cardinal numbers... By 1919, neither the combination “equivalence relation” nor “equivalence class” was in use.

Under the section Äquivalenzrelationen und Klasseneinteilungen of the first chapter of the first edition of Höhere Algebra, Hasse (1926) wrote:"We call such a decomposition a partition of $M$, and the subsets thereby determined its classes". (Hasse 1954, p. 22; Higher Algebra is the English translation of the third edition of Höhere Algebra published in 1933)

In the third edition of Mengenlehre (Hausdorff 1914; the third edition was published in 1937 and then as Set Theory in English in 1957), the term equivalence is still attached only to the context of cardinality. Even in 1942 and in a paper with the title Theory of Equivalence Relations, Oystein Ore called them “the blocks of the partition $P$" (Ore 1942,p.574).

Von Neumann (1926, 1929), Hopf (1930), and Seifert and Threlfall (1934) used the term “Äquivalenzklasse”, von Neumann (1936) “equivalence-class” (with a hyphen) and Solomon Lefschetz (1938, 1942) “equivalence class” (without a hyphen). The terminology was for denoting the classes constructed by certain contextualized equivalences, none of which was referred to as an “equivalence relation”. Moreover, none of these authors felt obliged to define “equivalence classes” in general terms and outside the contexts that they were working.

A foundational treatment is found in the first chapter of Tukey’s Convergence and Uniformity in Topology (Tukey 1940, p. 4), where the combinations “equivalence relation” and “equivalence class” appear in the same place... Generally, the combination “equivalence class” became common much later than the “equivalence relation” while closer to 1950 we can observe clear uses of both combinations alongside each other."


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