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A13.2 Function types and function application

Functional programming · A level · AQA 7517 4.12.1.1 · about 20 min

What this lesson is about

Functions as mappings, function type f: A to B, domain and co-domain, and function application with arguments from a Cartesian product.

• A function is a mapping
• Function type, domain and co-domain
• Writing the type in code
• Function application
• Applying a type to the robot

Questions 6 marks in all

1. [1 mark]A function has type f: A → B. What is B called?

   1. AThe co-domain
   2. BThe domain
   3. CThe argument type
   4. DThe Cartesian product
   Answer: A. A is the argument type, called the domain; B is the result type, called the co-domain.

2. [1 mark]is_even takes a whole number and returns True or False. What is its function type?

   1. Ais_even: integer → Boolean
   2. Bis_even: Boolean → integer
   3. Cis_even: integer × Boolean → integer
   4. Dis_even: integer → integer
   Answer: A. The argument type comes before the arrow and the result type after it.

3. [1 mark]Which statements about a function f: A → B are true?

   Tick every answer that is true.

   1. AA and B are subsets of objects in some data type
   2. BEach value in A is mapped to exactly one value in B
   3. CEvery value in B must be the result for some value in A
   4. DA is the set of results the function gives
   Answer: A, B. The co-domain is the set results are taken from; a function need not produce every value in it. The domain, not the co-domain, is the set of arguments.

4. [1 mark]add has type integer × integer → integer. What is integer × integer called?

   Answer: Cartesian product. integer × integer is the Cartesian product of the integers with itself: the set of all pairs of integers.

5. [1 mark]In function application, how many arguments does add(3, 4) strictly take, if add: integer × integer → integer?

   1. AOne, the pair (3, 4)
   2. BTwo, 3 and 4
   3. CNone, it is a constant
   4. DThree, 3, 4 and the result
   Answer: A. The domain is the Cartesian product integer × integer, so its single argument is a pair.

6. [1 mark]What does this program print?

   def compass(degrees):
       return "NESW"[((degrees + 45) % 360) // 90]
   
   print(compass(44), compass(45), compass(224), compass(315))

   Answer:

   N E S N

   Adding 45 and dividing by 90 gives the quarter: 44 is still N, 45 becomes E, 224 is S, and 315 wraps round to N.

The task: the compass type

# the two lines every program starts with: the commands, then the robot
from bugbot import *
connect()

def compass(degrees):
    return "N"

print("range:", compass(0))

# the two lines every program starts with: the commands, then the robot
from bugbot import *
connect()

def compass(degrees: int) -> str:
    return "NESW"[((degrees + 45) % 360) // 90]

results = set(map(compass, range(360)))
print("range:", " ".join(sorted(results)))

for i in range(4):
    turn_right(30, angle=90)
    print("facing", compass(round(heading()) % 360))