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Is there a real life example showing data that forms a relation is more useful than one that forms just a relation?

what real life scenario motivates the extra condition on relation so we have functions? Thank you.

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All of physics? Classical mechanics, for example -- the trajectory of a system is a smooth function from $\mathbb{R}$ to $\mathbb{R}^{3n}$. You can't get physics with relations. –  Neal May 16 '12 at 14:24

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There are tons of real-life examples of functions where a relation just does not make sense or, alternatively, makes things much more complicated.

Some examples:

  1. Citing Neal's example: all of physics. First example: The position of a mass point in motion. It is far more interesting to say that the point is at a well-defined position after a certain amount of time than listing "it might be here, here or here". Of course, if the position is not clear (as can happen in Quantum mechanics), just saying "it's here, here or here" is less interesting than giving a precise notion of how likely it is to be at a certain point - and this is, again, a function.
  2. Another physics example: A gravitational field assigns to each point in space a single number describing the gravitation potential of the point. Since this is a function, one may define the direction in which the potential decreases most strongly; if it were a relation, there could be multiple such directions, with the weird consequence that a body in motion could choose between different orbits at certain points in space.
  3. A computer science example: A function has the great property that, given its arguments, it yields a single value. This means that for computable functions, you can actually give a "direction" to functions, namely, given the values of the arguments, you can compute its value, yielding a simpler expression. For relations, this does not really make sense: If $R \subseteq A \times B$ is a relation, what is $R(a)$ for $a \in A$? (There are at least three possible solutions to this: Check $(a,b) \in R$ instead - this requires you to know or guess $b$, and is not "directed" in the sense above; return an arbitrary $b$ such that $(a,b) \in R$ - this reduces the problem to finding a choice function and loses information...; return a list of all possible $b$ - much more complicated...)
  4. From pure mathematics: Many important notions, such as homomorphism, continuous functions, injectivity, surjectivity, differentiability, whatever, don't make sense for relations or are much easier to define for functions.

It's quite easy to give lots of other examples as well.

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