Easy Ways to Memorize Surjections, Injections, Ranges, and Domains, Explained
Explanation
I found it really hard to memorize the names when I first encountered injective, surjective, and codomain. It was easier to distinguish them in English, but in Korean, it didn’t stick well. ‘Is this injective or surjective?’ ‘Was the codomain the larger one? What was it?’. Since the names sounded similar, I was always confused when I needed to use them. I don’t know how many people are like me, but even if there are only a few, I want to help clear the confusion. The reason the names are hard to memorize is because we try to memorize them blindly, but knowing and understanding the meaning of each word actually makes it harder to forget.
Let’s say a function is from $f$ to $f : X \rightarrow Y$. Then, at this time, $Y$ is called the codomain. And $\left\{ f(x) : x \in X \right\}$ is called the range.
The ‘range’ part in range means value 値. The range is the set containing all the function values. It’s called the range because it’s the area of the function ‘values’. The ‘domain’ part in codomain means one common 共. Here, it means ‘entire’. It means the entire domain.
When we talk about surjective and injective, the ‘ject’ part means to project or shoot 射. When we depict functions graphically, we often use arrows. In other words, it talks about how the elements of the domain connect to the range.
The ‘sur’ in surjective means complete 全. It is called surjective when the codomain and the range are the same. In other words, it means that the domain illuminates (or connects to) every part of the codomain.
The ‘in’ in injective means one 單. It is called injective when only one element of the domain corresponds to each element of the range. Another term for this is a one-to-one function. Hence, it’s called injective because it connects one by one.