Download WEEK 6: FUNCTIONS 1. Motivation Programs can be quite large. In

WEEK 6: FUNCTIONS
1. Motivation
Programs can be quite large. In order to design and maintain a large program, it
is desirable to break it down into more manageable parts. A well-designed program
will consist of several parts which can — to a large extent— be written and tested
separately. The concept of function is extremely useful in this respect.
2. The mathematical concept of function
Definition 2.1. A function consists of
(1) a set, say A, called the domain of the function,
(2) a set, say B, called the codomain of the function,
(3) a rule, say f , that associates with every element of A a unique element of
B.
Notation . We shall use the notation f : A → B for a function. Let x ∈ A, and let
y ∈ B be the element that the rule f associates with x. We shall write
y = f (x) .
The mathematical concept of function will be explored in some detail in the
Analysis part of the Core Mathematics B unit. For the present, it should suffice to
give a few examples.
Example 2.1. Let A be the set of all the students registered for this unit. Let B
be the set of the genders, i.e.
B = {MALE, FEMALE}
Define the rule by
f (x) = Gender of x .
Example 2.2. Let A = N and B = Q. Define the rule by
f (x) = x/2 .
Example 2.3. Let A = Z and B = Q. Define the rule by
f (x) = x/2 .
Example 2.4. Let A = N and B = R. Define the rule by
f (x) = x/2 .
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WEEK 6: FUNCTIONS
Let us consider the last three examples in greater depth. In a way, the rule f
appears to be the same in all three cases. Yet, according to Definition 2.1, each
example defines a different function.
• The function in Example 2.3 differs from that in Example 2.4 because the
domain is not the same.
• The function in Example 2.4 differs from those in Examples 2.2 and 2.3
because the codomain is not the same.
3. The function as a programming concept
Example 3.1. Consider the function sign : R → R defined by
(
1
if x ≥ 0
sign(x) =
−1 if x < 0
Implement this function in PYTHON.
Example 3.2. Let b ∈ {2, 3, . . .}. Every real number x ∈ (0, 1) can be expressed in
the form
∞
X
(E)
x=
an b−(n+1) ,
n=0
where the coefficients an take values in the set {0, 1, . . . , b − 1}. The series on the
right-hand side is called the base-b expansion of x. Write a function that takes
a floating-point number x and integers b and n as arguments, and returns a list
holding the coefficients a0 , . . . , an−1 in the expansion (E).
In summary, we note that the PYTHON concept of function is similar to the
mathematical concept. The main difference is that the type of the argument is
not specified explicitly in the interface of a PYTHON function. Hence the same
function may cater for arguments of many different types. Other programming
languages such as C++ are not so permissive.
4. Functions that return nothing!
Let a and b be two numbers. Suppose x = a and y = b. It is often useful in
programming to have a procedure for swapping the values of x and y. (For instance,
when sorting a set of numbers by size.) In PYTHON, the following code does the
job
temp = y
y = x
x = temp
After these three statements have been executed, the value of the variable x is b
and that of the variable y is a. If this procedure is to be used repeatedly in a
program, it is desirable to implement it as a “function”. This “function” would
return nothing; instead, it would carry out a specific task.
Example 4.1. Here’s a failed attempt to write a function to do this:
def swap(x,y):
temp = y
y = x
x = temp
WEEK 6: FUNCTIONS
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and here’s a PYTHON session that makes use of this function:
>>> u=-1
>>> v=1
>>> swap(u,v)
>>> u
-1
>>> v
1
>>>
Nothing happens! u and v have not been swapped. Read Cogliati, Chapter 7, for
the explanation. In brief, the variables x and y inside the function swap are copies
of the variables u and v respectively. Making changes to a copy does not affect the
original. How, then, does one get hold of the modified copies? One can’t; x and y
do not exist outside the function swap. So the function, as defined, is useless.
Example 4.2. Here’s a function that works:
def swap(listOfTwo):
temp = listOfTwo[1]
listOfTwo[1] = listOfTwo[0]
listOfTwo[0] = temp
This gives
>>> pair = [-1,1]
>>> pair
[-1, 1]
>>> swap(pair)
>>> pair
[1, -1]
>>>
The essential difference is that the argument of the function swap is now a list.
Lists are mutable objects, i.e. when a list is used as argument in a function, it is
(in some sense) the original that is passed— not a copy. So, although the name
used for the list inside the function (e.g. listOfTwo) may differ from the name
used outside (e.g. pair), both names refer to the same “address” in the computer
memory. The statements inside the function affect the list passed as argument.
5. Built-in functions, modules
Last week, we reviewed some of PYTHON’s built-in data types. Each built-in
data type comes with some functions that are available to the user.
Example 5.1. For the numeric types, there are a number of built-in functions, e.g.
abs,pow etc.
To find all the functions that are defined for, say, the floating-point type, we may
proceed as follows.
>>> x =1.0
>>> dir(x)
[’__abs__’, ’__add__’, ’__class__’, ’__cmp__’,
’__coerce__’, ’__delattr__’, ’__div__’, ’__divmod__’,
’__doc__’, ’__float__’, ’__floordiv__’, ’__getattribute__’,
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WEEK 6: FUNCTIONS
Name
syntax
abs
abs(x)
add
x+y
cmp
x<=y etc.
truediv x/y
mul
x*y
sub
x-y
Table 1. Some built-in
result
absolute value of x
sum of x and y
enables comparison of x and y
quotient of x and y
product of x and y
difference of x and y
functions for the floating-point type
’__getnewargs__’, ’__hash__’, ’__init__’, ’__int__’, ’__long__’,
’__mod__’, ’__mul__’, ’__neg__’, ’__new__’, ’__nonzero__’, ’__pos__’,
’__pow__’, ’__radd__’, ’__rdiv__’, ’__rdivmod__’, ’__reduce__’,
’__reduce_ex__’, ’__repr__’, ’__rfloordiv__’, ’__rmod__’, ’__rmul__’,
’__rpow__’, ’__rsub__’, ’__rtruediv__’, ’__setattr__’,
’__str__’, ’__sub__’, ’__truediv__’]
The functions listed are of the form
__nameOfFunction__
The underscores mean that the function is private; the user cannot view the definition of the function. The details of each function’s implementation are hidden
from the user, who must rely on the PYTHON documentation to learn what the
function does, and how it can be used. On the other hand, functions such as class
simply cannot be accessed by the user. We shall discuss that and other functions
next week.