📂Set Theory

Functions and Mappings Rigorously Defined by Set Theory, Sequences

Definitions ¹

Let’s assume two sets $X$, $Y$ that are not empty sets are given.

1. A binary relation $f \subset (X,Y)$ is called a function if it satisfies the following and is denoted as $f : X \to Y$. $$ (x ,y_{1}) \in f \land (x,y_{2}) \in f \implies y_{1} = y_{2} $$
2. For the function $f : X \to Y$, $\text{Dom} (f) = X$ is called the domain of $f$, and $Y$ is called the codomain of $f$. Given a subset $A \subset X$ of the domain, $f(A):= \left\{ f(x) \in Y \ | \ x \in A \right\}$ is called the image of $A$ under $f$. Especially, the image of the domain $X$ under $f$, $\operatorname{Im} f := f(X)$, is called the range of $f$.
3. A function whose domain is the set of natural numbers $\mathbb{N}$ is called a sequence.
4. The set of all functions that have the domain $A$ and the codomain $B$ is denoted as $B^{A}$.

Explanation

1. At the curriculum level, it’s said that ‘a correspondence $f : X \to Y$ from $X$ to $Y$ is a function if for every element $x_{1}, x_{2} \in X$, there exist $f(x_{1})$ and $f(x_{2})$ in $Y$ satisfying $x_{1} = x_{2} \implies f(x_{1}) = f(x_{2})$. However, this ‘correspondence’ or ‘mapping’ was somewhat ambiguous in expression. In mathematics beyond undergraduate levels, concepts related to functions are rigorously defined through set theory using the notion of relations. The explanation that a function yields only one output for an input is more suited to functions in computer science.
2. You may wonder why it’s necessary to define the range separately. For example, if we consider $f(x) = x^2$, it is obvious that the function values belong to $\left[ 0,\infty \right)$, and there seems no need to set it unnecessarily large like $f : \mathbb{R} \to \mathbb{R}$. Essentially, the range is just a subset of the codomain, and it’s hard to understand why we bother setting aside values that won’t be used in practice.
   This misunderstanding comes from thinking only of overly simple examples; not all functions can have their ranges easily predicted from the beginning. All that can be ensured when defining a function is that for $x \in X$, there exists a $f(x) \in Y$, but what that is, is unknown. If there is a complex function like $$ f(x) = \sin \ln \sqrt{x} + \int_{1}^{3^x} {{1} \over {7t+t^2}} dt $$ identifying its range from the beginning is not possible, nor necessary. Generally, knowing the range is important mainly when defining composite functions.
3. The definition of a sequence as simply an arrangement of numbers is simpler and more general, confirming it’s more than just counting numbers. Its codomain can be functions, any incredibly unique set, and this abstraction not only cleanly expresses the concept of sequences but also allows for flexible application in many mathematical fields dealing with infinity.
4. The concept of a set of functions might be unfamiliar, but in abstract mathematics, sets such as function spaces are routinely mentioned. Understanding why notations like $B^{A}$ are used becomes easier when recalling concepts like cardinals. For instance, considering all functions whose codomain is $B$ and domain is $A$, like $$ e \mapsto 1 \text{ or } 2 \text{ or } 3 \\ \pi \mapsto 1 \text{ or } 2 \text{ or } 3 $$ all possible combinations result in $9 = 3^2 = |B|^{|A|}$.