# Formalize a set theory argumentation from a short story fiction

This problem may be interesting. A writer Raymond Queneau wrote in his "Exercises in Style" a series of stories depicting the same event. One of them was in set theory. I'm wondering if anyone might be able and interested to formulate these events in standard set theory notation? This may be used as part of illustrative materials for a theatre project. Thanks a lot!

Set Theory

On the S bus, let us consider the set Ƨ of seated passengers and the set U of upright passengers. At a particular stop is located the set P of people that are waiting. Let C be the set of passengers that get on; this is a subset of P and is itself the union of the set Cʹ of passengers that remain on the platform and of the set C ̋ of those who go and sit down. Demonstrate that the set C ̋ is empty. H being the set of cool cats and {ɦ} the intersection of H and of Cʹ, reduced to a single element. Following the surjection of the feet of ɦonto those of y (any element of Cʹ that differs from ɦ), the yield is the set W of words pronounced by the element ɦ. Set C ̋ having become non-empty, demonstrate that it is composed of the single element ɦ.

Now let Pʹ equal the set of pedestrians to be found in front of the Gare Saint-Lazare, {ɦ, ɦʹ} the intersection of H and of Pʹ, B being the set of buttons on the overcoat belonging to ɦ, Bʹ the set of possible locations of said buttons according to ɦʹ, demonstrate that the injection of B into Bʹ is not a bijection.

—Raymond Queneau Translated by Chris Clarke (source)

• The crux is the formalization of the surjection of $z$'s feet onto $y$'s feet and the choice words that ensue. Incidentally, it strikes me as odd that the surjection doesn't go in the other direction, but that's how Queneau wrote it. – Fabio Somenzi Feb 26 '17 at 16:51

Let A be the set of seated passengers (voyageurs assis) and D those that are standing.

Lemma.
$A \not = \emptyset$.

Proof. That some passengers on the bus are standing is a consequence of the fact that this is the rush-hour (une heure d'affluence), and surely a result of the poor transportation service at that time. QED

Now if C is the set of people waiting in the bus-stop that were able to get on the bus, define $C' = C\cap D$ (namely those forced to stay standing on the platform) and $C^{''} = C \cap A$ (those who found a sitting place). We have to show that $C''=\emptyset$. Well, it follows from the axiom of selfishness that the platform passengers will not let those of C (the newcomers'') to sit unless they find a place for themselves. Hence trivially $C^{''} = \emptyset$.

Let $z$ be the unique cool-cat on the platform (zazou) and $y\in C'$ another passenger on the platform. Let $feet(p)$ denote the feet of $p$, and S the stepping relation. We have that $\langle feet(y), feet(z)\rangle\in S$, or in a more direct expression ''y steps on z''. This implies immediately a two-phase reaction by z. A set of words W followed by establishing $z\in C^{''}$ (namely z rushes to find a sitting place). Queneau concludes that $C^{''} = \{ z \}$ and asks for a proof of this statement. It is I believe an erroneous conclusion hastily made. The evidence that ''z abandonna rapidement la discussion pour se jeter sur une place devenue libre'' does not imply that other passengers were unable to find a place. (A counterexample is easy to build.)

Finally, the statement about the buttons is a direct consequence of the button hole principle.

• Great job, thank you! Indeed, a very thorough solution. :) – puslet88 Mar 14 '17 at 22:07