📂Set Theory

Injection, Surjection, Bijection, Inverse Function

Definition ¹

Let’s say $x \in X$, $y \in Y$, and $f: X \to Y$ are functions.

1. For every $x_{1}, x_{2} \in X$, if $x_{1} \ne x_{2} \implies f(x_{1}) \ne f(x_{2})$ then $f$ is called injective.
2. If $f(X) = Y$, then $f$ is called surjective.
3. If $f$ is both injective and surjective, it is called bijective.
4. $I : X \to X$ that satisfies $I(x) = x$ is called an Identity Function.
5. For every $x, y$, if $f(x) = y$ and $f^{-1} (y) = x$ is satisfied, then $f^{-1} : Y \to X$ is called the Inverse Function of $f$.

Basic Properties

• [1]: The identity function is bijective.
• [2]: $f$ being bijective is equivalent to the existence of inverse function $f^{-1}$.

Explanation

• Injective functions are also called one-to-one, or one-to-one functions.
• Surjective functions are also called onto.
• Bijective functions are also referred to as one-to-one correspondence.

In entrance exams, the concept of one-to-one correspondence might not seem very important, but it truly is. Many students struggling with math have either never heard of it or think it’s useless for problem-solving. Although not entirely incorrect, not knowing this concept likely means being unaware of other necessary problem-solving elements as well.

Even students who are relatively good at math might only truly understand the significance of bijective functions when they encounter university-level math. The notion of one-to-one correspondence is not limited to set theory but is a crucial concept in the vast domain of mathematics, regardless of the subject area. However, because of its powerful implications, paradoxically, mathematicians often research ways to relax the conditions for bijection. This includes methods to deduce bijection under different conditions or use functions as if they are bijective even when they are not.

To briefly illustrate its importance, it’s so critical that saying ‘it’s important to know correctly’ would be an understatement. Like it or not, bijections feature across various subjects, so not fully understanding them by the time of graduation is arguably more challenging.