# why function over just a relation?

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

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...)