Discrete Mathematics Function Checker

Overview

This project contains a Python script to determine whether a given function is injective (one-to-one), surjective (onto), or bijective (both one-to-one and onto). The script evaluates these properties based on a specified domain and codomain.

Functions Explained

Injective (One-to-One)

A function ( f: A \rightarrow B ) is injective if every element of the domain ( A ) maps to a unique element in the codomain ( B ). In other words, no two different elements in ( A ) map to the same element in ( B ).

Surjective (Onto)

A function ( f: A \rightarrow B ) is surjective if every element of the codomain ( B ) has at least one pre-image in the domain ( A ). In other words, the function covers the entire codomain.

Bijective

A function ( f: A \rightarrow B ) is bijective if it is both injective and surjective. This means every element in ( A ) maps to a unique element in ( B ) and every element in ( B ) is covered.

Code Description

The code provides a function IsInjective, IsSurjective, IsBijective to check whether a given function is bijective by evaluating its injective and surjective properties. Here's how the code works:

Functions and Methods

• IsBijective(func, dom, codom)
  • This is the main function that checks if the given function func is bijective.
  • It contains two helper functions:
    • is_one_to_one(): Checks if func is injective by ensuring no two elements in dom map to the same element in codom.
    • is_onto(): Checks if func is surjective by ensuring every element in codom is mapped by some element in dom.
  • If both conditions are met, IsBijective returns True, indicating the function is bijective.

Example Function

• f(x)
  • This is a sample function defined as ( f(x) = 2x ).
  • The domain and codomain for this example are [2, 4, 6, 8] and [4, 8, 12, 16], respectively.

Usage

To check if the function ( f(x) = 2x ) is bijective for the given domain and codomain, run the following code:

import numpy as np
import matplotlib.pyplot as plt

def IsBijective(func, dom, codom):
    def is_one_to_one():
        set = []
        for x in dom:
            value = func(x)
            if value in set:
                return False
            set.append(value)
            print(f"f({x}) = {value}")
        return True
    
    def is_onto():
        set = set()
        for x in dom:
            value = func(x)
            if value in codom:
                set.append(value)
                print(f"f({x}) = {value}")
            else:
                print(f"f({x}) = {value} (But it is not in the codomain)")
        return len(set) == len(codom)
    
    if is_one_to_one() and is_onto():
        return True
    else:
        return False

def f(x):
    return 2 * x

Domain = [2, 4, 6, 8]
Codomain = [4, 8, 12, 16]

result = IsBijective(f, Domain, Codomain)
print(f"Is the function bijective: {result}")