Introduction

Relevance of the Topic

A Bijective Function is one of the fundamental pillars of function theory and has intrinsic applications in numerous mathematical topics, besides being a prerequisite for the study of more advanced topics, such as set theory. This concept acts as a bridge between the basics of high school mathematics and the more abstract mathematics of higher education.

Contextualization

The Bijective Function emerges as an integral part of the sequence of topics covered in the 1st year high school mathematics curriculum, very close to the approach of Functions as a whole. It deepens the study of functions and demonstrates precisely how the inputs of a function are directly related to its outputs.

This topic not only solidifies students' mathematical foundation but also enhances their logical reasoning and problem-solving skills. It adds value to students' understanding of numerical relationships, so they can apply this knowledge not only in mathematics but also in other disciplines and in their daily lives.

Theoretical Development

Components

• Functions: We start with the understanding of what functions are. A function is a type of relation that associates each element of one set (domain) with a unique element of another set (codomain). We essentially identify functions as 'machines' that receive an input and produce an output.

• Domain and Codomain: We clarify the importance of the concepts of domain and codomain. The domain is the set of all possible inputs for a function, while the codomain is the set of all possible outputs of the function.

• Image and Pre-image: We introduce the concept of image and pre-image. The image of an element 'x' in the domain is the corresponding value in the codomain, denoted as 'f(x)'. The pre-image of an element 'y' in the codomain is any element 'x' in the domain, such that 'f(x) = y'.

• Injectivity and Surjectivity: We introduce the concepts of injective function (or injectivity) and surjective function (or surjectivity). The function is injective if each element of the domain is associated with a unique element in the codomain. The function is surjective if each element in the codomain has at least one associated element in the domain.

Key Terms

• Bijectivity: A special type of function that is simultaneously injective and surjective. In a bijective function, each element of the domain is associated with a unique and different element in the codomain and vice versa.

• One-to-One and Onto: This is the most 'casual' description of the terms injectivity (one-to-one) and surjectivity (onto). We can see that the function is one-to-one if each element in the domain is mapped to an element in the codomain. The function is onto if it covers (or 'reaches') all elements in the codomain.

• Inverse: Understanding the concept of inverse is crucial for understanding bijective functions. The inverse of a function f is denoted by f^-1 and has the property that f(f^-1(x)) = x for all x in the codomain. In other words, the inverse 'undoes' the original function.

Examples and Cases

• Example 1 - Enjoyable Function: Consider the function that maps numbers from 1 to 5 to words according to their length: {1 -> one, 2 -> two, 3 -> three, 4 -> four, 5 -> five}. This function is bijective because it uniquely associates each number (domain) with a word of the same length (codomain) and vice versa.

• Example 2 - Fashionista Function: Suppose we have a function that maps fruit names to their lengths in centimeters: {apple -> 5, banana -> 6, peach -> 7, grape -> 3}. This function is NOT bijective, as the word 'avocado', for example, does not have a corresponding value in the codomain.

• Example 3 - Fanatic Function: Now imagine a function that maps football teams to their places of origin: {Flamengo -> Rio de Janeiro, Corinthians -> São Paulo, Grêmio -> Porto Alegre, Santos -> São Paulo}. This function is bijective for uniquely associating each football team (domain) with its city of origin (codomain) and vice versa. Here, we can also see that the function is its own inverse: f(f^-1(team)) = team.

These examples illustrate the importance and practical application of bijective function concepts. They demonstrate that in a bijective function, each element of the domain has a unique and specific corresponding element in the codomain and vice versa.

Detailed Summary

Key Points

• Function Definition: Understanding that a function is a relation between two sets, where each element of the first set (domain) is associated with a unique and specific element in the second set (codomain), is crucial to understanding bijectivity.

• Injectivity and Surjectivity: Understanding the concepts of Injectivity and Surjectivity, which refer to how many elements of the domain are mapped to each element of the codomain and whether all elements of the codomain are mapped, respectively, is the first step to understanding bijective functions.

• Bijective Function: The central notion, a bijective function, or bijection, is one that is both injective and surjective. That is, each element of the domain is associated with a unique and different element in the codomain and vice versa.

• Inverse of a Bijective Function: Understanding the concept of the inverse of a function, represented by f^-1, where the roles of domain and codomain are swapped, is vital. In a bijective function, the inverse is again a bijective function, which demonstrates the 'reversibility' of the original function.

Conclusions

• The Bijective Function is a key concept in mathematics. Its thorough understanding not only solidifies the mathematical foundation but also enhances logical reasoning and problem-solving skills.

• The domain of a bijective function has a biunivocal correspondence (one-to-one) with its codomain, meaning each element of one set is associated with one and only one element of the other set.

• The bijective function is characterized by the existence of an inverse, which, when applied to the original function, results in the mathematical identity. This means that the bijective function can be 'undone' or 'reversed'.

Exercises

1. Exercise 1 - Analyze the function {1 -> a, 2 -> b, 3 -> c}, where the elements of the domain are mapped to the letters of the alphabet. Is this function a bijective function? If not, why not?

2. Exercise 2 - Find the inverse of the function {5 -> apple, 6 -> banana, 7 -> peach, 3 -> grape}. Verify if the inverse is a bijective function.

3. Exercise 3 - Consider the function that maps the names of capitals to their respective countries: {Paris -> France, Brasília -> Brazil, Buenos Aires -> Argentina}. Is this function a bijective function? Justify.