Function Types, First-Class Objects and Function Application

Overview

Recall questions on function types, domain and co-domain, functions as first-class objects and why that matters, and function application and the type of a call such as add(3, 4).

What you need to know

Answer each one before you reveal it.

Questions from this subtopic

Each one with the model answer underneath.

1. 1

   What is a function type, and how is it written?

   Model answer

   A function type records what kind of thing a function consumes and what kind of thing it produces, and nothing whatever about how the work is done. An arrow separates the two. For a function f, the notation f: A → B declares its type; whatever stands to the left of the arrow, here A, is the argument type, and whatever stands to the right, here B, is the result type. So sqrt: real → real takes a real number and hands back a real number, while length: string → integer takes a string and hands back a whole number. The type alone is enough to catch a mistake such as applying length to a number.

2. 2

   In the function type f: A → B, what are A and B called?

   Model answer

   A is called the domain and B is called the co-domain.

3. 3

   Define domain and co-domain.

   Model answer

   Every function is defined against two sets. The domain is the one its inputs must be drawn from — anything outside it may not be fed in. The co-domain is the one its outputs are guaranteed to fall within. Written f: A → B, the domain is A and the co-domain is B, so each argument must belong to A and each result must belong to B. For length: string → integer the domain is the set of all strings and the co-domain the set of integers; note that the co-domain need not be used up, since no string has a negative length.

4. 4

   Loosely speaking, what is a function?

   Model answer

   Loosely, it is a rule of assignment: take any member of one set — the inputs — and the rule pairs it with exactly one member of a second set, the outputs, though it need not reach every member of that second set. Consider f: {Monday, Tuesday, ..., Sunday} → {1, 2, ..., 7} under the rule that gives each day its position in the week, sending Monday to 1, Tuesday to 2 and so on. Its domain is the set of weekday names and its co-domain the whole numbers from 1 to 7, and this particular rule does happen to use every one of them.

5. 5

   Is it necessary for every member of the co-domain to be an output of the function? Explain.

   Model answer

   No. Not all of the co-domain's members need be outputs. The co-domain simply states the set the outputs are drawn from; it does not promise that every member of it actually occurs. For example square: integer → integer has the whole set of integers as its co-domain, but no negative integer is ever produced, and neither is 2 or 3. The values that genuinely do occur — sometimes called the range or image — may be a proper subset of the co-domain. By contrast, every member of the domain must have an output, because a function assigns a result to each element of its domain.

6. 6

   What is always true about the domain and co-domain of a function?

   Model answer

   They are always subsets of objects in some data type. The domain and co-domain are not arbitrary collections: each is drawn from a data type such as integer, real, character, string or boolean, and may be the whole of that type or a restricted part of it. For example a function might have the integers from 1 to 12 as its domain, which is a subset of the objects of type integer.

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