# Tuple definition

Is it correct?

$S=\{\langle t,h\rangle:t\in\{0,\Delta t,2\Delta t,\cdots,24\},h\in\{0,\Delta h,2\Delta h,\cdots,H\}\}$

I would like to say that $S$ is a 2-tuple. The first tuple can vary from $0$ to $24$, with $\Delta T$ step, and the second one can vary from $0$ to $H$, with $\Delta H$ step.

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As described, $S$ is a set of ordered pairs ($2$-tuples), not a $2$-tuple. A bag of apples is not an apple. – Arturo Magidin Jun 22 '12 at 20:40
Have you checked the old stand-by for definitions? en.wikipedia.org/wiki/Tuple – cobaltduck Jun 22 '12 at 20:50
$S$ is a set of $2$-tuples, since each of its elements is a $2$-tuple. However, in English it's more common to say that it's a set of ordered pairs. – talmid Jun 22 '12 at 21:02
The set of tuples you have described is a Cartesian product of those sets within the definition, so it would be easier to say $S = \{\ldots\} \times \{\ldots\}$. – dtldarek Jun 22 '12 at 21:10

$S$ is not a tuple, it's a set of $2$-tuples.
I think what you want to say is that $S$ is a set of ordered pairs ($2$-tuples), each of which has a first entry that can vary from $0$ to $24$ in steps of length $\Delta t$; and second entry that can vary from $0$ to $H$ in steps of length $\Delta h$. As written, it would only make sense if $24$ is evenly divisible by $\Delta t$ and $H$ by $\Delta h$; that is, if and only if there exist integers $k$ and $\ell$ such that $24 = k\Delta t$ and $H=\ell\Delta h$.
If the latter is not the case, then I would say that the first entry will take the values $k\Delta t$ with $k=0,1,2,\ldots,\lfloor\frac{24}{\Delta t}\rfloor$, and the second entry will take the values $\ell\Delta h$ with $\ell=0,1,2,\ldots,\lfloor\frac{H}{\Delta h}\rfloor$.
Note, however, that $S$ is a set of tuples (not a tuple), and it is the first/second entry of the elements of $S$ that we are describing, not a "first tuple" and "second tuple": the elements of $S$ are themselves not ordered, so it doesn't make sense to talk about a "first tuple" and a "second tuple", when $S$ is a set.
OK, I've mixed up the definition between tuple and ordered pair. Let me try again: Given $S=(t,h)$, a set of ordered pairs, which first entry can vary from 0 to 24 in steps of length $\Delta T$, second entry can vary from 0 to H in steps of length $\Delta H$ and both intervals can be divisible by $\Delta T$ and $\Delta H$, respectively. How can I describe mathematically it? – Paulo Fracasso Jun 23 '12 at 5:48
@Paulo: If $S=(t,h)$, then $S$ is an ordered pair, not a set of ordered pairs. Your original notation was (more or less) fine. You can refer to $S$ as a set of ordered pairs, but you don't say "which first entry". The simplest way to say what you want to say is: "$S$ is a set of ordered pairs of the form $(t,h)$, where $t$ varies from $0$ to $24$ in steps of length $\Delta T$ and $h$ varies from $0$ to $H$ in steps of length $h$." To denote it mathematically, you can write $$S=\{(t\Delta T, h\Delta H)\mid t=0,1,\ldots,\frac{24}{\Delta T}, h=0,1,\ldots,\frac{H}{\Delta H}\}.$$ (cont) – Arturo Magidin Jun 23 '12 at 18:22
@PauloFracasso: (cont) or: "$S=A\times B$, where $A=\{0,\Delta T, 2\Delta T,\ldots,24\}$ and $B=\{0,\Delta H, 2\Delta H,\ldots, H\}$". Don't think you need to use "which", "respectively", and other such words to make it "mathematical". – Arturo Magidin Jun 23 '12 at 18:23