Mdthbs

Here is a table of the Staten Island subway:

| STOP | STATION         |

|------+-----------------|

|    1 | St. George      |

|    2 | Tomkinsville    |

|    3 | Stapleton       |

|    4 | Clifton         |

|    5 | Grasmere        |

|    6 | Old Town        |

|    7 | Dongan Hills    |

|    8 | Jefferson       |

|    9 | Grant           |

|   10 | New Dorp        |

|   11 | Oakwood Heights |

|   12 | Bay Terrace     |

|   13 | Great Kills     |

|   14 | Eltingville     |

|   15 | Annadale        |

|   16 | Hugenot         |

|   17 | Prince’s Bay    |

|   18 | Pleasant Plains |

|   19 | Richmond Valley |

|   20 | Arthur Kill     |

|   21 | Tottenville     |

and here is an Elisp function that maps a number to a name:

(defun sisubstops (s)

  (pcase s

    ((pred (or (< s 1) (> s 21))) (message "Hey, just stops 1 through 21, please!"))

    ('1 (message "St. George"))

    ('2 (message "" Tomkinsville))    

    ('3 (message "Stapleton"))

    ('4 (message " Clifton"))

    ('5 (message " Grasmere"))       

    ('6 (message " Old Town"))       

    ('7 (message " Dongan Hills"))

    ('8 (message " Jefferson"))

    ('9 (message " Grant"))     

    ('10 (message " New Dorp"))

    ('11 (message " Oakwood Heights"))

    ('12 (message " Bay Terrace"))   

    ('13 (message " Great Kills"))   

    ('14 (message " Eltingville"))

    ('15 (message " Annadale"))       

    ('16 (message " Hugenot"))  

    ('17 (message " Prince’s Bay"))

    ('18 (message " Pleasant Plains"))

    ('19 (message " Richmond Valley"))

    ('20 (message " Arthur Kill"))   

    ('21 (message " Tottenville"))))

So, when I'm trying to tell students that this is a function of a basically non-continuous phenomenon, i.e., we're not dealing with a function like f(x) = x^2, rather a function that matches stops to names -- but it is still a function -- how can I get this across. One idea would be to got deep into Lambda Calculus and talk about how LC does conditionals. My first goal is to distinguish between continuous versus discrete; and yet I've written a Lisp function that can't really be represented as any sort of normal math function e.g., S(n) = M*nwhere M is some sort of machine or constant that takes a stop number and turns it into a stop name. Any ideas how, yes, this is a function? Or is this just a misnomer, i.e., comp-sci using/abusing the term function. So yes, how is sisubstops really a function?

In math, a function is just an association between each element in a set X and a specific element in set Y. X and Y can be any well defined set: real numbers, rational numbers, integers, even integers, odd integers, all the real numbers except 0, words, names of states, names of cities, etc. Calling a table mapping between between bus stop numbers and bus stop names a function is perfectly sound mathematically as long as no single stop maps to two different names.

Focus on the idea that the sets can be finite. They don't need to be defined by a formula. Any set of ordered pairs is a relation. If there is a unique second element for any given first element then it is a function.

You don't need Lambda Calculus or any mathematics to show what a function is. Finite functions can be computed by table lookup or by hash maps.

Actually functions defined by formulas in computing are often problematic since computations of many values (say Real Values) are only approximate.

Talk about step functions. This might get the point across that functions do not need to be continuous. (I think going into lambda calculus would just confuse your students here - there's no need.)

As for whether this is really a function - sure it is! To make it clearer, here's something you can do. The written numbers we have - '7', for instance - just stand in for the concept of seven. We could really use the symbol 'a' to stand in for the concept of seven. (Just as we can use any variable name to hold the value inside - though we do tend to choose variable names that 'make sense' to us.)

With that in mind, what keeps us from saying that 'St. George' is just another symbol? Nothing. The symbols don't matter, the relation between them does - that's what makes a function a function.