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How do we write a function as a relation?: In this thread the top answer has written a function as a relation as "$aRf(a)$" where I assume we can just define $b=f(a)$?

  • I'm sure this is correct and makes sense but it bothers me because: With other relations e.g. < we have $_1R_2$ as $1<2$. The relation goes in the middle i.e. the analogue here would be $_2f_4$.
  • I assume $_2f_4$ makes no sense. But I find It weird that the function $f$ is both the relation $R$ and also the output $f(a)$.
  • Perhaps the answer is, that I need to mentally separate $f$ and $f(a)$ where the latter is actually just $b$?

Quick thoughts welcome!

CormJack
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    It is just a matter of notation. Denoting $_2f_4$ makes perfect sense, if you define it. This is called infix notation. I would say it is uncommon, but if you declare it, it is perfectly valid. People seem to like more Euler's notation $f(2)=4$. A "set theoretic notation" can also be convenient $(2,4)\in f$. – plop Jan 06 '23 at 18:37
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    Does this answer your question? Is a function a special kind of relation? – Anne Bauval Jan 06 '23 at 18:38
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    Functions (using ordered pair notation) are relations, just that the independent variable is not assigned to more than one value. $f$ might be the name of the function, but it must also come along with a domain and codomain. $f(a)$ is a shorthand notation for what $a$ is related to. – David P Jan 06 '23 at 18:38
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    And yes, strictly speaking, in Euler's notation $f$ is the (name of the) function and $f(b)$ is its value (the unique value $c$ such that $(b,c)\in f$ or $_bf_c$). However, like in common language people sometimes abuse it. If the letter $b$ is read as a variable then some people might write $f(b)$ to refer to the function, and not the value. You see that more often with $x$ instead of $b$. People say "the function $f(x)$". The context matters, to resolve ambiguities of the language. – plop Jan 06 '23 at 18:46
  • Hi all I didn't expect this to get such helpful or positive responses, thank you for taking the time! – CormJack Jan 07 '23 at 09:40
  • @owl that is very useful to hear thank you! So I could write seething like $_xf_y$ where $f(x) = x^2$, and that would be defining the relation, donated $_xf_y$? And your comments about both $f$ and $f(x)$ referring to the function is also a good additional note. Perhaps it explains my underlying/subconscious concern. – CormJack Jan 07 '23 at 09:44
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    @AnneBauval thank you for posting that link. It's very helpful/interesting, but doesn't answer my question. However there is an answer at the bottom which alludes to my question, but it has 0 votes, so im glad have the thought's here for reference as well! Thanks for helping. – CormJack Jan 07 '23 at 09:45
  • Hi David, that makes sense to me, and your last line suggests that the notation $aR{f(a)}$ is also acceptable. Thanks! – CormJack Jan 07 '23 at 09:46
  • @owl I'd be happy to vote your answer correct if you wanted to format it into a proper answer? I don't know if you care about the reputation points etc. Thanks! – CormJack Jan 07 '23 at 09:49
  • The accepted answer in the proposed duplicate explains (in its 3rd paragraph) why a function $f:A\to B$ is a particular kind of binary relation $R:$ for every $a\in A$, there should be exactly one $b\in B$ such that $aRb$ (which you can write $_af_b,$ why not?), and this is what makes the more usual formula $b=f(a)$ legit. – Anne Bauval Jan 07 '23 at 09:57
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    Hi @AnneBauval thanks again. I just reviewed it again closely, it is helpful, but my questions wasn't about whether a function can be considered a relation. This I already knew, it was specifically about the exact notation I mention. Which the answer doesn't comment on, and only the 0 voted answer at the very bottom does. So you know I'm not bs ing you, below is a link to a much longer question of which this questions is a part of (I've broken my long question up into mini question this one is Q3 from the lone one). So it really was just about that very specific notation. – CormJack Jan 07 '23 at 10:05
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    I know it seems unlikely to you, but my question really was about just the (why not?) you have in brackets! Here's a link to the long one: https://math.stackexchange.com/questions/4597979/relations-transitive-functions-idempotent-functions – CormJack Jan 07 '23 at 10:06
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    And in case you're interested here's another one (again from the original): https://math.stackexchange.com/questions/4612899/when-is-a-function-f-transitive – CormJack Jan 07 '23 at 10:07
  • Since you are trying to understand what is a function and its notation, it is probably good for you to note that "a relation with the property of unique images for all elements of the domain" is not the only definition of function that is used in mathematics. A function defined that way, has no knowledge of which set is its entire co-domain. For example, the concept of surjective doesn't make any sense for them. For that reason, in some places a function is the "relation with unique images, but together with the co-domain". – plop Jan 07 '23 at 14:28
  • To make things worse, the latter definition is sometimes used tacitly, but that happens with less serious authors, like calculus textbooks. More careful authors might dedicate two different names for the two concepts, like function and map or morphism, respectively. – plop Jan 07 '23 at 14:31
  • Hi @owl im really enjoying your commitment to my question! All great insights, and thanks for tying different definitions together for me. Is there any chance I can tempt you to look at any of my other posts hahah? There is one that is really bugging me atm. Let me know, thanks! (Also like I said, if you wanted to collect your comments into an answer I'd happily vote it, so you get the points etc)! – CormJack Jan 07 '23 at 14:33

1 Answers1

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Every relation $R$ with the domain $A$ and the codomain $B$ is a subset of the cross-product $A\times B$. If $a\in A, b\in B$ and the pair $(a,b)\in R$ You can write this either as $R(a,b)$ or as $aRb$.

Every function is a relation. If in a relation $F$ for every $a\in A$ there is exactly one (i.e. not 0 and not 2 or more) $b\in B$, then we call the relation $F$ a "function." A good example is the identity $I$ or $=$. This is a typical function. You can write is as

$$ I(a,a)\quad\mathrm{or\,as}\quad=(a,a) $$

or as

$$ aIa\quad\mathrm{or\,as}\quad a=a $$

Since it is a function, you additionally can use the function notation

$$ f(a)=a $$

Hubert Schölnast
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