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Bijective Functions

Python 3.8

def is_function(domain, codomain, function):
    # Check that the image of every element in the domain is defined.
    for element in domain:
        # Check if there exists an image of the element
        if not any(pair[0] == element for pair in function):
            # If the image of element does not exist, return False 
            return False
    
    # Check if each element of the domain has only one image
    preimages = set()
    for pair in function:
        if pair[0] in preimages:
            return False
        preimages.add(pair[0])
    
    # Check that the range is a subset of the codomain. 
    for pair in function: # pair = (element, image)
        # Check if every element has an image in the codomain 
        if pair[1] not in codomain:
            # If the image of any element is not in the codomain, return False
            return False  
    return True
def is_injective(domain, codomain, function):
    # Check if every domain element has a unique image.
    image = set()
    for pair in function:
        if pair[1] in image:
            # A codmain element is an image of more than one domain elements.
            return False
        image.add(pair[1])    
    return True
def is_surjective(domain, codomain, function):
    # Check if every element of the codomain has a preimage.
    for element in codomain:
        if not any(pair[1] == element for pair in function):
            return False
    # If the function is surjective
    return True
def is_bijective(domain, codomain, function):
    return is_injective(domain, codomain, function) and is_surjective(domain, codomain, function)
# Define the domain
domain = {1, 2, 3, 4, 5}
# Define the codomain
codomain = {6, 7, 8, 9, 10}
# Define the function
f = {(1, 7), (2, 6), (3, 9), (4, 8), (5, 10)}
#check if f is a valid function
if is_function(domain, codomain, f):
    #check if f is a bijective function
    if is_bijective(domain, codomain, f):
        print('The function is bijective.')
    else:
        print('The function is not bijective.')
else:
    print('Use a valid function.')

Code explanation

Lines 1–22: We define a is_function function that takes three arguments, i.e., domain, codomain, and function. It checks if the argument function is a valid function with the domain and the codomain. It returns False if function is not a valid function.
Lines 24–32: We define a is_injective function that takes three arguments, i.e., domain, codomain, and function. It checks if the passed function is injective by checking if every domain element has a unique image. It returns True if the function is injective and returns False otherwise.
Lines 34–41: We define a is_surjective function that takes three arguments, i.e., domain, codomain, and function. It checks if the passed function is surjective or not by checking if every codomain element has a preimage. It returns True if the function is surjective and returns False otherwise.
Lines 43–44: We define a is_bijective function that takes three arguments, i.e., domain, codomain, and function. It checks if the passed function is bijective.
Lines 46–50: We define the sets domain, codomain, and f.
Lines 52–59: We check if f is a valid function from the domain to the codomain, and if it is, we check if f is a bijective function. We print messages according to each situation.

Types of functions

A function is always one of the four types listed below:

Neither injective nor surjective
Only injective but not surjective
Only surjective and not injective
Both injective and surjective, which means bijective

Let’s look at a few examples of functions showing each of these cases. These functions are defined using the following sets:

Bijective Functions

Bijective functions

Examples

Checking if a function is bijective using Python

Code explanation

Types of functions