I'm a college student, just beginning to study Elementary Set Theory. In studying about the definition of injective and surjective function, out of curiosity, it came to my mind a question about the definition these functions:
1) How did they came up with such definition? How did it came about & why must it be defined that way? What is so special about injective & surjective function that makes them has to be defined in such a way?
To make clear the context of my question, here are the conditions of this question:
- "In short" injective functions are defined as: for every element in the codomain, there is at "most" one element that maps to it from the domain. For Surjective functions: for every element in the codomain, there is at "least" one element that maps to it from the domain.
- As you can see, i'm not seeking about what exactly the definition of an Injective or Surjective function is (a lot of sites provide that information just from googling), but rather about why is it defined that way? What makes it so special that it has to be given it's own terminology (injective & Surjective functions)?
- Lastly, i'm just a human so if something is not clear about this post, just let me know in the comment & i'll try to edit the post to make it clearer.
I don't know how the real story goes and this is just a rough speculation in trying to guess how it came about (it is possible to be something along this line but don't see it as the real truth though):
- The person who first coined these terms (surjective & injective functions) was, at first, trying to study about functions (in terms of set theory) & what conditions made them invertible. He observed that some functions are easily invertible ("bijective function") while some are not possible to have any kind of inverse. As he continue to study, he discovers that some functions can have inverses if certain conditions are met, as if they were "semi" invertible type of functions. He discovers that these conditions can be distuingish into two types: right invertible or left invertible. Later on he can distuingish that, only those function who maps with a specific type of pattern can have a left inverse (which he called the pattern "injective"). And a similar thing happened as he discovered the pattern for surjective functions. So in short my opinion is: is not like he wanted to define injective & surjective function like so, but it just happened that he "discovered it" in "that form". Hence, the definition of injective & Surjective function. But, again, this is just me trying to be Mr. Know it all. I hope I'm not being too rude. If anyone do know the real story, do share.