0
$\begingroup$

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

  • 0
    @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.2011-03-29

1 Answers 1

1

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 $x_1$.