📂Set Theory

Injection, Surjection, Bijection, Inverse Function

Definition ¹

Let $x \in X$, $y \in Y$, and let $f: X \to Y$ be a function.

1. If $x_{1} \ne x_{2} \implies f(x_{1}) \ne f(x_{2})$ for all $x_{1}, x_{2} \in X$, then $f$ is called injective.
2. If $f(X) = Y$, then $f$ is called surjective.
3. If $f$ is both injective and surjective, then it is called bijective.
4. $I : X \to X$ satisfying $I(x) = x$ is called the identity function.
5. $f^{-1} : Y \to X$ satisfying $f(x) = y$ and $f^{-1} (y) = x$ for all $x, y$ 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 the inverse function $f^{-1}$.

Explanation

• An injection is also called one-to-one, or a one-to-one function.
• A surjection is also called onto.
• A bijection is also called a one-to-one correspondence.

One-to-one correspondence is a concept that is considered really unimportant in entrance-exam mathematics, yet it is extremely important. Many students who struggle with mathematics think they have never even heard of this fact, or, even if they have heard of it, regard it as useless for solving problems. This is not entirely wrong, but if one does not know even this much, there is a high chance they do not know well the things needed for problem solving either.

Even fairly capable students often only truly take bijection into their body once they get through high school and encounter undergraduate-level mathematics. One-to-one correspondence is the most important concept not only in set theory but also throughout the vast world of mathematics, regardless of which subject it belongs to. Yet, precisely because it is such a powerful and desirable condition, paradoxically, mathematicians end up researching in the direction of relaxing the condition of bijection. Such as how one can derive bijectivity under other conditions, or whether something can be used like a bijection even though it is not actually a bijection.

To roughly explain how important it is, there is no need even to say 'it is really important, so you must know it precisely.' Whether you like it or not, bijection appears in all sorts of subjects, so on the contrary it would be harder to graduate without knowing bijection precisely.