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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.

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  • $\begingroup$ To give us some more context, what is the subject matter of the exam you are preparing for? $\endgroup$ – Rob Arthan Feb 21 '20 at 0:42
  • $\begingroup$ Nature of my entrance exam is based on aptitude, but few subliminal concepts in mathematics entice me to think in detail. Clearing those concepts before hand is more useful in long run. I have added a comment under 1st answer by nomen for further inquiry. $\endgroup$ – Ubi hatt Feb 21 '20 at 1:33
  • $\begingroup$ Maybe am just being nosey now, but what is the subject of your exam: philosophy, mathematics, general knowledge, or what? (Nosey or not, I don't see how to begin to answer your question without that extra background.) $\endgroup$ – Rob Arthan Feb 21 '20 at 1:41
  • $\begingroup$ @RobArthan Mathematics. To be precise discrete mathematics for computer science. But I am encapsulating these information because this question is just a personal inquiry. $\endgroup$ – Ubi hatt Feb 21 '20 at 1:44
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    $\begingroup$ The universe is "made of" relations: e.g. "father-son" relation. Some are functions (e.g. "the father of") while some are not ("the son of"). Why mathematics "prefer" functions ? Because functions models operations: take an input and produce/compute an output. Operations are sort of relations where the single-valued requirement is necessary: we want to be sure about the result produced by the operation: the result of a sum, the total of our bank account, the GDP of our country. $\endgroup$ – Mauro ALLEGRANZA Feb 21 '20 at 8:57
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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.)

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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.

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  • $\begingroup$ I don't see the point of your example: a many-many relation does not have to depend on "what the [values in the range] are"? $\endgroup$ – Rob Arthan Feb 21 '20 at 0:45
  • $\begingroup$ You must make a choice, and you get ONE prize. The other prizes are irrelevant to what you get. Your mistake is somehow parsing "You only get one prize" as "this is a many to many relationship" $\endgroup$ – nomen Feb 21 '20 at 0:46
  • $\begingroup$ But what is the relevance of the phrase "not on what all the prizes are" in your answer? Adding extra constraints on a functional relation can't make it non-functional. I am trying to understand how your example is pertinent to the question: you've just given an example of a functional relation with some extraneous detail. $\endgroup$ – Rob Arthan Feb 21 '20 at 0:55
  • $\begingroup$ Thank you! I understand your example and the relation (R) that you established between Player (domain) and Prize (Co domain) i.e. R = relation of choice. Once you make a choice, you don't get to select other Prizes... $\endgroup$ – Ubi hatt Feb 21 '20 at 1:28
  • $\begingroup$ ...But, I am more concern over the general ideal here, i.e What Function can model that Relation can't? There are certain graphs which intersects x-axis in two distinct points for value of Y and expression which does that we don't consider it to be a function? Your answer is based on definition of Function (which I know well). I am looking for philosophical or to be precise the fabric of mathematical realm that why only specific relations (functions) are studied so much in detail in mathematics. $\endgroup$ – Ubi hatt Feb 21 '20 at 1:28
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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.

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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?)

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  • $\begingroup$ Yes, I know the definition of the function. I don't want to repeat my comment again. Can you please read my comment in other answer. It also applies for your answer. Thank you! $\endgroup$ – Ubi hatt Feb 21 '20 at 1:29
  • $\begingroup$ OK, so take @guy3141's scalar field example. What would modeling the scalar field as a relation give you that functions already don't? The possibility of discussing multiple temperatures at a point? OK, but you can already do that by considering sets of functions or functions to sets. And you can semi-trivially translate between all of these formalisms. $\endgroup$ – nomen Feb 21 '20 at 16:36

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