Learning objective
Function type: Know that a function, f, has a function type f: A → B (where the type is A → B, A is the argument type, and B is the result type). Know that A is called the domain and B is called the co-domain. Know that the domain and co-domain are always subsets of objects in some data type. Loosely speaking, a function is a rule that, for each element in some set A of inputs, assigns an output chosen from set B, but without necessarily using every member of B. For example, f: {a,b,c,…z} → {0,1,2,…,25} could use the rule that maps a to 0, b to 1, and so on, using all values which are members of set B. The domain is a set from which the function’s input values are chosen. The co-domain is a set from which the function’s output values are chosen. Not all of the co-domain’s members need to be outputs.
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Functional programming paradigm
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Function type
Aqa A Level Computer ScienceFundamentals of functional programming
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Quick explanation
Function type: Know that a function, f, has a function type f: A → B (where the type is A → B, A is the argument type, and B is the result type). Know that A is called the domain and B is called the co-domain. Know that the domain and co-domain are always subsets of objects in some data type. Loosely speaking, a function is a rule that, for each element in some set A of inputs, assigns an output chosen from set B, but without necessarily using every member of B. For example, f: {a,b,c,…z} → {0,1,2,…,25} could use the rule that maps a to 0, b to 1, and so on, using all values which are members of set B. The domain is a set from which the function’s input values are chosen. The co-domain is a set from which the function’s output values are chosen. Not all of the co-domain’s members need to be outputs
- This point belongs to Functional programming paradigm, especially Function type.
- You need to be able to function type: Know that a function, f, has a function type f: A → B (where the type is A → B, A is the argument type, and B is the result type). Know that A is called the domain and B is called the co-domain. Know that the domain and co-domain are always subsets of objects in some data type. Loosely speaking, a function is a rule that, for each element in some set A of inputs, assigns an output chosen from set B, but without necessarily using every member of B. For example, f: {a,b,c,…z} → {0,1,2,…,25} could use the rule that maps a to 0, b to 1, and so on, using all values which are members of set B. The domain is a set from which the function’s input values are chosen. The co-domain is a set from which the function’s output values are chosen. Not all of the co-domain’s members need to be outputs.
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Why it matters
This objective helps connect Function type to exam-style questions, flashcards, and revision notes for Functional programming paradigm.
Quick student answer
For a function f: A → B, what does A represent?
Direct answer
The domain
Key terms
- Domain: The set from which a function's input values are chosen; in f: A → B, it is A.
- Co-domain: The set from which a function's output values are chosen; in f: A → B, it is B. Not every co-domain member needs to be an output.
Common trap
Assuming every co-domain member must be an output: The co-domain is the set from which outputs are chosen. It may contain members that the function never uses.
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Related learning objectives
- First-class object: Know that a function is a first-class object in functional programming languages and in imperative programming languages that support such objects. This means that it can be an argument to another function as well as the result of a function call. First-class objects (or values) are objects which may: • appear in expressions • be assigned to a variable • be assigned as arguments • be returned in function calls. For example, integers, floating-point values, characters and strings are first class objects in many programming languages.
First-class object
- Function application: Know that function application means a function applied to its arguments. The process of giving particular inputs to a function is called function application, for example add(3,4) represents the application of the function add to integer arguments 3 and 4. The type of the function is f: integer x integer → integer where integer x integer is the Cartesian product of the set integer with itself. Although we would say that function f takes two arguments, in fact it takes only one argument, which is a pair, for example (3,4).
Function application
- Partial function application: Know what is meant by partial function application for one, two and three argument functions and be able to use the notations shown opposite. The function add takes two integers as arguments and gives an integer as a result. Viewed as follows in the partial function application scheme: add: integer → (integer → integer) add 4 returns a function which when applied to another integer adds 4 to that integer. The brackets may be dropped so function add becomes add: integer → integer → integer The function add is now viewed as taking one argument after another and returning a result of data type integer. 92
Partial function application
- Composition of functions: Know what is meant by composition of functions. The operation functional composition combines two functions to get a new function. Given two functions f: A → B g: B → C function g ○ f, called the composition of g and f, is a function whose domain is A and co-domain is C. If the domain and co-domains of f and g are ℝ, and f(x) = (x + 2) and g(y) = y3. Then g ○ f = (x + 2)3 f is applied first and then g is applied to the result returned by f.
Composition of functions