# 3 Mathematically speaking

Mathematicians have rigorous ways of defining things like maps. As Wolfram MathWorld explains:

A map is a way of associating unique objects to every element in a given set. So a map ƒ:A|->B from A to B is a function ƒ such that for every a ∈ A, there is a unique object ƒ(a) ∈ B.

What does *that* mean? Say you have a bunch of things — a set, in mathematical terms — and you want to put them someplace or refer to them in a more convenient way. Call that set A. You need to *map* (verb) each element, or object, in set A to some other element in another set, which we’ll call B. And each object in A can only go one place. So, as Wolfram says, we need to associate unique elements in B with each element in A.

Those of you who liked high school algebra may note that this is the same as the definition of a function. In fact, mathematically speaking, a map and a function are the same thing. For the rest of you, here’s an example.

## A classroom example

Consider the students in your class — a set of students. Each student needs a desk — a unique desk; one student can’t sit at two desks, although in a pinch they could share — and so you could *map* (verb) students to desks by creating a seating chart. Your seating chart would be the map (noun). You didn’t know that when you assigned seats you were writing a mathematical function, but there you go: Algebra in practice!

That’s a simple example. The original data set, your set of students, has just one variable, the student’s identity or name; and the set of desks can simply be numbered. You might say it’s one-dimensional.

Student | Desk |
---|---|

Anastasia | 11 |

Billy Bob | 1 |

Candi | 5 |

Dora | 3 |

Elwin | 8 |

Fiona | 9 |

Grover | 2 |

Hargrove | 4 |

Ingrid | 12 |

Josepha | 6 |

Kaga | 10 |

Lerlene | 7 |

Of course, this is assuming you’ve numbered your seats — you’ve created a second map, by mapping the physical desks in your classroom to the whole numbers 1 to 12. If you actually want your students to be able to *find* their seats, though, you may want to do more than simply number them. You may want to map your list of students to the two-dimensional space of your classroom. (I say *two*-dimensional because I assume you don’t have a mezzanine. If you do, hang on, we’ll get to three dimensions later on.)

For simplicity, let’s say you have a simple rectangular arrangement of three rows of four desks each, numbered this way:

Desk | Row | Column |
---|---|---|

1 | 1 | 1 |

2 | 1 | 2 |

3 | 1 | 3 |

4 | 1 | 4 |

5 | 2 | 1 |

6 | 2 | 2 |

7 | 2 | 3 |

8 | 2 | 4 |

9 | 3 | 1 |

10 | 3 | 2 |

11 | 3 | 3 |

12 | 3 | 4 |

Or, representing this visually:

Then you can give students this map of their seating assignments, representing students by their first initials:^{2}

Now this is starting to look like what most people think of as a map, which is a two-dimensional representation of physical space. Actually “a representation of a space” is a pretty good way to define a map more generally. A *space*, mathematically speaking, doesn’t have to be a physical space. It can be a *coordinate space*, which is the imaginary space defined by the axes of a graph. Remember Cartesian coordinates from high school algebra?

Here you can see the “space” defined by the x and y axes. But of course we can map a physical space to coordinate space — that’s what we do when we make a map of the world or of a geographical region.

You could add a coordinate grid to your seating chart, too, but that’s probably overkill — unless you were teaching in a lecture hall, in which case it might be useful.

All this to tell students how to get to their seats!