I gather that the symbols $\subset$ and $\supset$ were introduced by Ernst Schröder in his 1890 Vorlesungen über die Algebra der Logik. This account also appears—attributed to good old Cajori—in an answer on the Mathematics SE site along with a quotation from Peano’s 1889 Arithmetices Principia Nova Methodo Exposita explaining use of a left-right-reflected upper-case C to mean contained in (or, as he put it in the Latin original, continetur).

It seems safe to hypothesize that Peano chose the reflected C because C is the first letter of continetur. Indeed, I recall being taught in grade school in the 1960s that such was the motivation for the symbol. But Nathan Moore, though a fine fifth-grade teacher, cannot be presumed to have been an authority on either the history of mathematical notation or the influences of Latin—through Norman French—on the lexicon of Modern English. And anyway, maybe my memory has corrupted itself over the past half century and all he really said was that the resemblance between $\subset$ and “C for contained in” can serve as a handy mnemonic. But back to Peano. Even though the hypothesis

Peano’s choice stemmed from continetur’s starting with C

does seem reasonable, that apparent reasonableness does not constitute sufficient grounds for asserting that the hypothesis is veridical.

Apparently relevant too is the fact that in the same work, Peano also uses $\cap$ and $\cup$ as we do today, but defines them as et and vel, respectively (in English, and and or). Although I relied in grade school on the anglophone-obvious mnemonic “$\cup$ is for union,” the Peano-reasonable hypothesis (analogous to “mirrored C is for continetur”) is “$\cup$ is for vel.”

But here’s a major wrinkle that sings out for anyone working in order theory. There are enchanting parallels between $\subset$ and $<$. Consider:

  • We denote the reflexive closure of $\subset$ by $\subseteq$. Likewise, we denote the reflexive closure of $<$ by $\leq$.
  • $\leq$ is the canonical, the prototypical partial order on $\bf R$. Likewise, the canonical partial order on any power set $2^S$ is $\subseteq$.
  • We denote the order duals of $<$ and $\leq$ by the left-right-reflected symbols $>$ and $\geq$, respectively. Likewise, the duals of $\subset$ and $\subseteq$ we denote by the reflections $\supset$ and $\supseteq$, respectively.
  • Of the symbols $<$ and $>$, we use the convex-left one for the relation of increasing magnitude: $(x,y)\in <$ means that that $x$ is the number with the lesser magnitude. Likewise, we use the convex-left $\subset$ for the relation of increasing cardinality: $(X,Y)\in\subset$ means that $X$ is the set with the lesser cardinality. This is consistent with the way that the notational similarity between $0$ and $\emptyset$ reflects the mathematical similarity between the objects that those two symbols represent.

I note in passing that because of all these order-theoretic parallels, I am among the authors who use $\subset$ only to mean the reflexive reduction of $\subseteq$, which is to say, $\subsetneq$. Of course, that usage of $\subset$ is not the more common, so I always point it out explicitly. And sometimes I simply use $\subsetneq$ instead.

But more importantly, it seems to me that the parallels are just too good to have happened entirely by accident. Indeed, in his Earliest Uses of Symbols of Set Theory and Logic, Tom Craven states that before Schröder introduced $\subset$ et al., “the symbols $<$ and $>$ had been used." Craven may have found this in Cajori, to which I don’t have access.

So, was Schröder influenced by Peano’s reflected C for continetur? And more generally, what can we say for certain about all the considerations that did lead Schröder to his choice… and, perhaps to different degrees, led to his notation’s now universal acceptance?


1 Answer 1


That Schröder was influenced by Peano is very unlikely. When introducing his inclusion symbol in Vorlesungen, Schröder explicitly cites $<$ for numbers in arithmetic as motivation, and his version looks more like rounded $<$ than c. He mentions Du Bois-Reymond's $≺$ and Peirce's $−\!\!\!<$ (Peirce was also Schröder's main source for the content of Vorlesungen), but not Peano. Tou, Math Origins: The Logical Symbols is a useful survey that has photocopies of text from original sources.

Moreover, while Peano's symbol, unlike Schröder's, looks more like the modern $\subset$, its meaning is reversed compared to both modern and Schröder's. Peano's $a\,⊃\,b$ for classes $a$ and $b$ means 'every $a$ is $b$'. This is in line with de Morgan's X)Y used for the same purpose earlier, and Peano's own use of $\,⊃\,$ also for implication. The latter was adopted by Russell and Whitehead in Principia and became common until eventually and slowly displaced by Hilbert's arrow.

Why the modified Schröder's symbol prevailed is hard to say. The similarity with c shape might have helped. Gergonne derived an inclusion symbol from c shape long before Peano, and it probably appealed to people after. Hausdorff's adoption in 1914 Grundzüge der Mengenlehre, the first textbook on set theory, helped too. His symbol still looked like Schröder's rounded $<$. One can speculate that it was later uniformized to the shape of implication for typographic reasons.

As for $<$, it goes back to 17th century and was derived from rather murky and quaint considerations by Harriot's editors, see Tanner, On the Role of Equality and Inequality in the History of Mathematics and History of greater-than symbol used in reverse? here. Cajori's History Of Mathematical Notations, vol. I is freely available on Internet Archive, and documents a long history of experimentation with symbols from 17th to 20th century, with all sorts of shapes and motivations for them.

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    $\begingroup$ This nails my question! That Schröder explains his motivation is more than I could have expected. As for Cajori’s being available on line, I am nearly embarrassed that it hadn’t occurred to me to check; I just assumed it was, I dunno, still under copyright. $\endgroup$ Nov 15, 2023 at 12:56
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    $\begingroup$ And inspired by the lesson of Cajori’s availability, I have now gone on to find Schröder’s Verlesungen too. His exact words in the relevant passage are “Das Zeichen ist ähnlich gestaltet, gewissermassen nachgebildet dem (einen) „Ungleichheitszeichen“ der Arithmetik, nämlich dem Zeichen < für „kleiner (als)“. This means “The sign is designed similarly to—and to a certain extent modeled on—one of the inequality signs from arithmetic, namely the symbol $<$ for less than’.” Life is good! $\endgroup$ Nov 15, 2023 at 13:47

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