Why we require to study a specific type of relation called function?

Question

I am recently preparing for a competitive exam and I am revising concepts again. During preparation, I mused over the following idea and got confused.

Firstly, sorry for being too verbose, and Secondly, I know my question is naive in nature. Moving from Relations to Functions, I wondered that why we required a concept of function, if we already have concept of relations. Why we require a specific type of relation in which for every single element of domain there should be a single element in co-domain? i.e. why can't co-domain have two values for single value in domain. Why we define all mathematical expressions (equations) in terms of functions?

Why we require to study a specific type of relation called function?

I know 'how' of the question i.e. How we can move from relation to function? But, I am confusion over the philosophy or mathematical requirement over the idea of Functions. Hence , I can't figure out 'why' of the question i.e. Why we require to move from relation to function to study Mathematical Function?

Final words, we study a specific relation which fulfills surjection, injection, and bi-jection, but strictly can't have two distinct values of domain having same value in co-domain. I can't image the wholesome image of requirement of function to study Mathematical Functions.

Answer 1

A function is a relation which models the "production", "construction", or choosing of a value based on the "original" value (i.e., the function's argument).

A generic relation cannot do that for us, since it can have arbitrarily many elements for the same "input" (for concreteness, say, the first element of the relevant ordered pair).

Consider this example: you're at a game show, where you are given the choice between prize 1, prize 2, and prize 3. You only get one prize. The value of your prize depends on what you choose, not on what all the prizes are.

Answer 2

Another way to think about it is that a function is a rule that assigns one and only one value to a given input.

As an example, if the inputs are points in a room and the function outputs the temperature at the given point we can only make sense of temperature being single valued 'ie a function'. (would it make sense to say a point has two distinct temperatures?)

Answer 3

Even before learning about relations, we have an intuitive concept of "operations", like squaring a number, measuring the height of a tree, or locating a city on a map. In order to study operations like these, we want a formal definition that captures the property of being well-defined: for every "input" value, there's exactly one associated "result". The standard definition of a function (as a special kind of relation, as you've described) does this.

Since you're also asking about injections and surjections, it sounds like you might have a broad question like "why do we define and study things that are special cases of other things we've already studied?". Consider the analogous question "why do we have the word 'bird' and a study of birds instead of just a general study of animals?" - the answer is that many useful facts are true of all birds that are not true of all animals. Similarly, many useful facts are true of functions that are not true of all relations, and many useful facts are true of injective functions that are not true of all functions.

By the way, if you're thinking about operations that can produce multiple results (like looking up the phone numbers of all the Karls in your city), note that such an operation can be seen as a normal function whose codomain consists of sets. In fact, in this way, every binary relation on a set $S$ corresponds to a function $f:S\to \mathcal P(S)$ (where $\mathcal P(S)$ is the set of all subsets of $S$), and this correspondence is itself a bijective function from $\mathcal P(S\times S)$ (the set of all binary relations on $S$) to $\mathcal P(S)^S$ (the set of all functions $S\to\mathcal P(S)$). The statement that it's a bijection expresses the fact that it's a "perfect one-to-one correspondence". (I know that's a little abstract, but it's an example of how these concepts arise naturally.)

Answer 4

It seems that the notion of the function is as primitive as it gets. For instance, when we describe a relation we implicitly use a function from the set of ordered pairs of objects to the set of truth-values. So we must understand the concept of the function even before we formally define it in terms of relations. The idea is that the concepts of the function and relation are on the same level of primitiveness, so one shouldn't be viewed as a special case of the other. Saying that a function is a specific type of relation is just a handy way to introduce fewer undefined terms.

Something similar happens when we define natural numbers in terms of sets. It is unlikely that one would primarily think of a natural number as some kind of set, but being able to formalize them that way may be very useful.

---

To sum up, we want to study this particular class of relations because they are good at representing what we intuitively understand by functions. So good, in fact, that we are willing to declare it to be the definition of functions.