Contextualization

Mathematics is the universal language that allows us to precisely describe the phenomena around us. The function is one of the most powerful tools within this language. It connects a set of values called the 'domain' to another set called the 'codomain' through a 'rule' that defines which elements of one set are related to which elements of the other. Functions are everywhere, from the exchange relationship in a purchase, to how light reflects in a mirror, to the operation of artificial intelligence algorithms.

Among the various types of functions, one of the most interesting is the bijective function. The bijective function is a function that is both injective and surjective. The injective function is one in which different elements of the domain are always related to different elements of the codomain. The surjective function is one in which every element of the codomain is related to at least one element of the domain. In other words, a bijective function is a function that has a one-to-one correspondence between the domain and the codomain, without leaving any element 'left over' in either of the sets.

Introduction

In this project, we will explore and understand the importance of bijective functions, their characteristics, and applicabilities. A bijective function has the remarkable property of being 'invertible', which means that if we know the function's rule, we can go from the domain to the codomain and vice versa. This has many applications, such as encoding and decoding messages, creating efficient computer algorithms, analyzing social networks, among others.

Bijective functions are essential tools in many branches of mathematics, such as algebra, geometry, number theory, and graph theory. Therefore, understanding the characteristics and properties of bijective functions is a crucial step to deepen your studies in mathematics.

The bijective function is one of the foundations of Cryptography, a branch of applied mathematics that studies the principles and techniques that transform a message into code, in order to allow only the legitimate recipient to decipher it. This is essential for the security of online transactions, such as purchases, bank transfers, protected emails, and others.

Practical Activity: 'Bijective Encoding'

Project Objective

Unveil the world of cryptography with the help of bijective functions. By the end of this project, students should be able to understand how bijective functions are used to encode and decode messages, and build a simple encryption algorithm using a bijective function-based cipher system.

Project Description

Students will be divided into groups of 3 to 5 people. Each group will be tasked with creating a message encoding and decoding system using a bijective function. The groups will have one month to complete the project and should dedicate five to ten hours per participating student to execute it.

Required Materials

• Paper, pen (for initial sketches and planning)
• Computer with internet access (for research and algorithm implementation)
• Text editor or programming software (to implement the algorithm)

Step by Step

1. Theoretical Study: All group members should study and understand the concept of bijective functions. Each group member will prepare a brief explanation to ensure everyone understands the topic. The provided bibliography can be used for this purpose.

2. Planning: Groups should plan and sketch their encoding and decoding system. They must clearly define the bijective function to be used and how this function will be implemented for encoding and decoding.

3. Implementation: Groups should implement their encoding and decoding system. They can use any programming language or software they prefer, as long as they can demonstrate the system's operation.

4. Testing: Each group will encode and decode several messages to ensure that the system works as expected. Testing is a crucial part to verify if the bijective function was implemented correctly.

5. Final Report: Groups should compile a detailed report of the project, which should include an introduction with theoretical contextualization, project development, conclusions, and the bibliography used.

Project Deliverables

Report

The report should follow the format of a scientific report, with four main sections: Introduction, Development, Conclusions, and Bibliography.

1. Introduction: Describe the relevance of bijective functions and their application in message encoding and decoding. Explain the overall project objective.

2. Development: Detail the theory behind the bijective function and how the group used this theory to develop the encoding and decoding system. Describe the methodology used, step by step, how the project was carried out, and the results obtained.

3. Conclusions: Conclude the work by summarizing its main points, the learnings obtained during the project, and your conclusions about the application of the bijective function in cryptography.

4. Bibliography: Indicate all sources that were used for the project, such as books, web pages, videos, among others. Use the appropriate format for each type of source.

Presentation

In addition to the report, groups should give a project presentation to the class. The presentation should last between 10 to 15 minutes and should cover the same points as the report, but in a more concise and visual manner. Groups should demonstrate the operation of the encoding and decoding system during the presentation.