# Construct a new relation from several relations

Let $f_0,\dots,f_{n-1}$ are relations, where $f_i$ is a relation of arity $m_i$ for every $i=0,\dots,n-1$.

How to construct an $(m_0+\dots+m_{n-1})$-ary relation from them?

For my previous erroneous attempt to do this see here: Name for product of relations

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 If all on same set, can define new relation as holding if all the component relations hold, or if at least one holds, or any other desired Boolean combination. – André Nicolas Jun 9 '12 at 15:09 So, you want to combine several sets of n-tuples into a single set? There is, of course, no standard way to do this. Do you have an application in mind? – Dan Christensen Jun 15 '12 at 15:24

$$\left\{ \operatorname{uncurry}z \hspace{0.5em} | \hspace{0.5em} z \in \prod f \right\}$$
where uncurrying is defined in Wikipedia: http://en.wikipedia.org/wiki/Currying by the formula $\operatorname{uncurry}(f)(x;y)=(fx)y$.