3

I've read this definition for an hour now and I cannot piece it together abstractly.

To define an n-tuple as a function $F$, where $X$ is the index set and domain, and $Y$ is the set containing the elements of the tuple: $$(a_1,a_2, ...,a_n) \equiv (X, Y, F)$$ $X= \{1,2,...,n\}$
$Y= \{a_1,a_2,...,n\}$
$F= \{(1,a_1),(2,a_2),...,(n,a_n)\}$

However, $$(a_1,a_2,...,n) := (F(1),F(2),...,F(n))$$ is a lot more comprehensible. It might be the notation in the previous definition that throws me off, or maybe there is something more fundamental that I cannot see or recall.

user222071
  • 31
  • I agree that your definition is much more approachable; a function $f: [n] \to Y$ is a sequence. I'm not sure exactly what your question is, however. – pjs36 Mar 09 '15 at 22:43
  • An explanation of the first definition. More of a request, though. I'm not familiar with the (X, Y, F) expression either. – user222071 Mar 09 '15 at 22:47

1 Answers1

3

It sounds like you're OK thinking of a sequence $(a_1, a_2, \ldots, a_n)$ of elements of some set, $Y$, as a function.

Explicitly, given that our sequence should consist of $n$ elements of $Y$, let $X$ denote the set $\{1, 2, \ldots, n\}$. Then any function

\begin{align*}F: X &\to Y \\ k &\mapsto a_k\end{align*} defines a sequence that we can write, like you pointed out, $\left(f(1), f(2), \ldots, f(n)\right) = (a_1, a_2, \ldots, a_n)$.

By writing the whole thing as a triple $(X, Y, F)$, they're just listing the domain, the range, and the function $F$ from the domain to the range.

As far as writing $F = \{(1, F(1)), (2, F(2)), \ldots, (n, F(n))\}$, that's just thinking of a function as ordered pairs, with the first coordinate from the domain, and the second from the range.

I think that's the big idea: just think of the ordered triple $(X, Y, F)$ as $($domain, range, function$)$. You already know we need that data, they're just being very formal about it.

pjs36
  • 17,979