Description

Introduction

In this lab we continue to use Python functions to implement some simple concepts related to functions.

As in the previous assignment, we will restrict our focus to finite domains and codomains. Assuming A is the domain and B is the codomain, we continue to use subsets of A × B to represent a function f : A → B, owing to the fact that a function is a special type of relation between A and B. For example if A = {dog,cat,fish} and B = {1,2} the following function:

f(dog) = 1                                                   f(cat) = 2 could be represented in python as follows: >>> A = = {’dog’,’cat’,’fish’} >>> B = {1,2} >>> f= {(’cat’, 2), (’dog’, 1), (’fish’, 1)}

f(fish) = 1

Note that the above code introduces a new syntax to create sets where you list elements between { and }. Last week we instead created sets from a list of elements. Both options are useful. The following code prints out the function by iterating through the set f

>>> for i in f:

>>>              print(i)

Assuming functions are represented using the structure described above, write Python functions to implement the following requirements.

1

1. Let A and B be finite domains and codomains. Two functions f : A → B and g : A → B are said to be equal if f(a) = g(a) for each element a ∈ A. Note that this definition extends also to the case when A and B are infinite sets. Write a python function called equal functions that takes as input two functions (call them f and g) and returned True if they are equal and False otherwise.
2. Recall the definition of partial function given in class. Write a function is partial function that takes as input the domain A and a function f and returns True if the function is partial and False otherwise.
3. Recall the definition of composite function. If g : A → B and f : B → C, then the composition of f and g is a function f ◦ g : A → C defined as (f ◦ g)(a) = f(g(a)). Write a python function composite function that accepts as input two functions g and f and returns the composite function f ◦ g as per the above definition. The composite function should be represented as a set, using the same format described above. To keep things simple, you can assume that g and f are well formed, i.e., that the composite function exists. However, if you want to challenge yourself, you can consider the more complex case where you have to check whether the composite function is defined. If it is defined, then return the corresponsing set, and if it is not defined return the empty set. How could it be that a composite function is not defined? Consider for example the following case:

>>>g = {(’cat’, 2), (’dog’, 1), (’fish’, 1)} >>>f = {(1,True),(0,False)}

In this case the composite function f ◦ g is not defined because g(cat) = 2, but f(2) is not defined.

While solving the above questions, keep in mind that when you iterate over a set in Python, you should not assume a specific order because it is not defined (i.e., there is no first, second, third element, etc.)