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I recently remembered the definition of a distance function between two points as being defined as $d((x,y),(z,w))$

Is it possible to define functions that take two separate mathematical objects? Why do we not then do this for real functions of two variables, defining them individually over two real numbers instead of a tuple?

user37577
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    Define things however you like... do what seems natural. Whether we are writing it as $d((x,y),(z,w))$ or as $d(((x,y),(z,w)))$ or as $d(x,y,z,w)$... whether we have the domain as $X^2\times X^2$ or as $(X\times X)\times (X\times X)$ or as $X^4$... we can usually freely switch between interpretations at a whim based on what is convenient at the time and whether we are wishing to emphasize some particular property of what it is we are doing – JMoravitz Sep 15 '22 at 14:37
  • Not very clear, but I imagine that you are considering the distance between points in $\mathbb R^2$. If so, the issue is: what is the problem? – Mauro ALLEGRANZA Sep 15 '22 at 14:38
  • @MauroALLEGRANZA I think the complaint is that the OP has only ever seen functional notation used with a single argument before and they are calling into question whether a function may accept multiple arguments rather than in the form of accepting a single argument in the form of a tuple. E.g. the difference between coding as main(String arg1, String arg2) versus main(String[] args) – JMoravitz Sep 15 '22 at 14:39
  • @JMoravitz precisely, my only question is, why don't we do this for real arguments too?My understanding was writing f(x,y) is just ignoring the extra parenthesis in $f((x,y))$ – user37577 Sep 15 '22 at 14:41
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    Again, do things however you like and you are comfortable with. Both are acceptable. If you are ever pressed on the topic, know how to formally define things... but anyone who would be in a position to press you on it should also be more than capable to recognize how you can define things in that way instead (or has already done so or seen it done before) and so should have no qualms about the distinction. – JMoravitz Sep 15 '22 at 14:43
  • @JMoravitz useful thanks. – user37577 Sep 15 '22 at 15:59
  • But the issue is that two points in the 2-D plane mean two couples of coordinates. If we unpack them into four real numbers, how to correctly manage them? – Mauro ALLEGRANZA Sep 15 '22 at 16:08
  • @MauroALLEGRANZA from context. – JMoravitz Sep 15 '22 at 16:13
  • @MauroALLEGRANZA that is true, I guess it's a case that it's much easier and nicer to use $P=(x,y)$ $q=(z,w)$ and then we can define $d(p,q)$ makes sense why they do it this way. – user37577 Sep 15 '22 at 16:34

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