# Formulation of functions & sets

I am trying to formulate in a document some work done. However I have a problem defining the following. If I have a problem defined as the tuple $\langle X,Y \rangle$ such that $(x,(y_{11},y_{12},y_{13},\ldots,y_{1N}))$. How can I define this relation in cartesian products?

For example if X = {$x_{1},x_{2},x_{3}$} and N = 2 then Y = {{$y_{11},y_{12}$}, {$y_{21},y_{22}$}, {$y_{31},y_{32}$}} and should define the pairs ($x_{1}$, ($y_{11},y_{12}$)), ($x_{2}$, ($y_{21},y_{22}$)), ($x_{3}$, ($y_{31},y_{32}$)).

Thank you

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## migrated from cstheory.stackexchange.comMar 29 '11 at 8:30

This question came from our site for theoretical computer scientists and researchers in related fields.

@Ahmed: This site addresses research-level questions in theoretical computer science. I can migrate your question to math.SE, where you may get an answer. Shall I do that? – Dave Clarke Mar 29 '11 at 8:17
@Dave Yes and thank you for clearing that up :) – 3ashmawy Mar 29 '11 at 8:29
Your notation suggests that $X \in \mathcal{X}$ and $Y \in \mathcal{Y}^N = \mathcal{Y} \times \cdots \times \mathcal{Y}$ ($N$ times) so that $\langle X, Y \rangle \in \mathcal{X} \times \mathcal{Y}^N$. Without further information it seems impossible to say more. – t.b. Mar 29 '11 at 8:59
@Theo I added an example above to elaborate what I mean. I dont think your suggestion would suffice in that context. Please correct me if you think otherwise. – 3ashmawy Mar 29 '11 at 10:33

As t.b. points out, $X\in\mathcal{X}$ and $Y\in\mathcal{Y}^N=\underbrace{\mathcal{Y}\times\mathcal{Y}\times\cdots\times\mathcal{Y}}_{N\text{ times}}$.
A relation between sets $A$ and $B$ is defined as a subset of $A\times B$. Hence in this case the desired relation $R\subseteq\mathcal{X}\times \mathcal{Y}^N$. This models exactly the pairs given in the example.
The only problem is that you have $\mathcal{Y}^2=\{\{y_{11},y_{12}\},\{y_{21},y_{22}\},\{y_{31},y_{32}\}\}$ where you really ought to have $\mathcal{Y}^2=\{(y_{11},y_{12}),(y_{21},y_{22}),(y_{31},y_{32})\}$. Curly braces normally denote unordered tuples, while parentheses denote ordered tuples. You need $(y_{11},y_{12})$ to be ordered, because they are ordered upon combining with $x_1$.