Question

What does the statement everyone can fit in a bus mean?

Answer

It depends on context.

Question

What is context?

Answer

A change in the likely meaning of an ambiguous word or situation based on other nearby words or situations.

Question

How do we know what a given context means?

Answer

Some meaning may be intrinsic (this part has to mean x or that part wouldn’t make sense), but much is cultural – that is, based on likely-to-be-shared past experience (I usually see x, so that probably holds here)

For example, my culture leads me to expect everyone can fit in a ____ to mean

Everyone can fit in a	Suggests (to me)
car	Partition: several cars, more car seats than people
costume	Any can fit each: costumes are baggy enough for any build
auditorium	Any can fit the set: each auditorium has more seats than we have people

However, that interpretation is not intrinsic in the words: your experience may lead you to understand these differently.

Question

How can I communicate clearly if I don’t share a culture?

Answer

Learn about other cultures.

Or, if everyone you communicate with has time, learn a common jargon: special in-discipline words or symbols that have clearly defined meaning. Jargon lets me communicate across cultures reliably, but also creates a barrier to entry. With English I can communicate easily with people raised like me; with jargon I can communicate precisely with people educated like me.

CS has its own jargon, mostly based on redefining words and symbols from English (e.g. else) and math (e.g. =). It also uses some jargon from mathematics and formal logic.

What follows is an effort to illustrate one specific phrase and how context changes its meaning, together with examples of how these might be expressed using the jargon of discrete mathematics.

1 Partition

Everyone can fit in a bus by dividing people between buses

This is formalized in mathematics as a partition problem: given a set of buses and a set of people, find a mapping between people and buses that satisfied capacity constraints.

Partitions are an important enough concept we have specific jargon for them, but that jargon will not be covered in this course. However, that jargon is defined in terms of discrete mathematics, so we can still express this using this courses’ content, although the result is rather complicated:

Let B be the set of buses, P be the set of people, and c : B \rightarrow \mathbb N be a function giving the capacity of a bus. Then this case is \color{blue} \exists f : B \rightarrow \mathcal P(P) \;.\; \color{black} \Big( \color{red} \forall b \in B \;.\; \color{orange} \big|f(b)\big| \le c(b) \color{black} \Big) \color{green} \land \color{black} \Big( \color{magenta} \forall p \in P \;.\; \color{gray} \exists b \in B \;.\; \color{brown} p \in f(b) \color{black} \Big)

That is, there’s some mapping from buses to sets of people such that both (a) the number of people mapped from each bus is within the bus’s capacity and (b) every person is in the set mapped to by some bus.

2 Any can fit the set

Everyone can fit in a bus, so we only need one bus.

This is formalized in mathematics by saying the fit in a bus predicate applies to an entire set of people rather than to individual people.

Let B be the set of buses, P be the set of people, and f(x,y) be a predicate asserting x can fit in y. Then this case is \forall b \in B \;.\; f(P,b)

3 Any can fit each

Everyone can fit in a bus, even the largest person in the world.

This is formalized in mathematics by saying the fit in a bus predicate applies to any person we happen to pick, but individually instead of as a group.

Let B be the set of buses, P be the set of people, and f(x,y) be a predicate asserting x can fit in y. Then this case is \forall p \in P, b \in B \;.\; f(p,b)

4 One can fit the set

Everyone can fit in a bus, so we only need one of the big buses.

When we say a bus we mean one bus, but do we mean a special specific bus or any arbitrary bus we could pick? In math, we distingusih these two ideas with different symbols: \forall x means no matter which x we pick and \exists x means it is possible to pick the right x.

Let B be the set of buses, P be the set of people, and f(x,y) be a predicate asserting x can fit in y. Then this case is \exists b \in B \;.\; f(P,b)

5 One can fits each

Everyone can fit in a bus; even the largest person in the world can fit on a big bus

When we say a bus we mean one bus, but do we mean a special specific bus or any arbitrary bus we could pick? In math, we distingusih these two ideas with different symbols: \forall x means no matter which x we pick and \exists x means it is possible to pick the right x.

Let B be the set of buses, P be the set of people, and f(x,y) be a predicate asserting x can fit in y. Then this case is \exists b \in B \;.\; \forall p \in P \;.\; f(p,b)

6 Each has one that can fit it

Everyone can fit in a bus; there are buses with high doors and ceilings for tall people, buses with wide doors and aisles for wide people, and so on.

Is there one specific bus that any person can fit into, or might there be different buses for differently shaped people? These two ideas are distinguished by the order of the \forall and \exists symbol: the left-most one applies to the entire expression that follows.

Let B be the set of buses, P be the set of people, and f(x,y) be a predicate asserting x can fit in y. Then this case is \forall p \in P \;.\; \exists b \in B \;.\; f(p,b)