Simple set theory problem - definition of a J-Tuple (1 Viewer)

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On page 113 Munkres (Topology: Second Edition) defines a J-tuple as follows:

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I was somewhat perplexed when I tried to completely understand the function [itex] \ x \ : \ J \to X [/itex].

I tried to write down some specific and concrete examples but still could not see exactly how the function would work.

For example if [itex] J = \{1, 2, 3 \} [/itex] and X was just the collection of all the letters of the alphabet i.e.

[itex] X = \{ a, b, c, ... \ ... \ z \} [/itex] then ...

... obviously a map like 1 --> a, 2 --> d, 3 --> h does not work as the intention, I would imagine is to have a mapping which specifies a number of triples ... but how would this work?

Can someone either correct my example or give a specific concrete example that works.

Would appreciate some help.

Peter
 

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Let's say ##X=\mathbb{R}##. The usual definition of a ##3##-tuple is just ##(x_1,x_2,x_3)## where each ##x_i\in \mathbb{R}##. For example ##(1,1,1)## and ##(2,1,2)## are ##3##-tuples.

This definition sees every ##3##-tuple as a function ##f:\{1,2,3\}\rightarrow \mathbb{R}##. Indeed, the ##3##-tuple ##(x_1,x_2,x_3)## is presented as the function ##f## such that ##f(k) = x_k##. So for example, the ##3##-tuple ##(1,1,1)## is the constant function ##f(k)=1##, while ##(2,1,2)## is presented by the function ##f(1) = 2##, ##f(2)=1##, ##f(3) = 2##.

This generalizes beyond ##\{1,2,3\}## of course. If you're familiar with a sequence in ##\mathbb{R}##, then you know that this is just a function ##\mathbb{N}\rightarrow \mathbb{R}##. This is thus simply an ##\mathbb{N}##-tuple.
 
Let's say ##X=\mathbb{R}##. The usual definition of a ##3##-tuple is just ##(x_1,x_2,x_3)## where each ##x_i\in \mathbb{R}##. For example ##(1,1,1)## and ##(2,1,2)## are ##3##-tuples.

This definition sees every ##3##-tuple as a function ##f:\{1,2,3\}\rightarrow \mathbb{R}##. Indeed, the ##3##-tuple ##(x_1,x_2,x_3)## is presented as the function ##f## such that ##f(k) = x_k##. So for example, the ##3##-tuple ##(1,1,1)## is the constant function ##f(k)=1##, while ##(2,1,2)## is presented by the function ##f(1) = 2##, ##f(2)=1##, ##f(3) = 2##.

This generalizes beyond ##\{1,2,3\}## of course. If you're familiar with a sequence in ##\mathbb{R}##, then you know that this is just a function ##\mathbb{N}\rightarrow \mathbb{R}##. This is thus simply an ##\mathbb{N}##-tuple.
Thanks micromass, been through your post carefully ... just to be sure ...

You write

"This definition sees every ##3##-tuple as a function ##f:\{1,2,3\}\rightarrow \mathbb{R}##."

... ... so in defining a set of triples or 3-tuples we are dealing with a set of functions, one function for every 3-tuple ... is that correct?

(Mind you I guess it is as Munkres was defining a (one) J-tuple ... rather than a function that described a set f J-tuples ,,,)

Peter
 
Thanks micromass, been through your post carefully ... just to be sure ...

You write

"This definition sees every ##3##-tuple as a function ##f:\{1,2,3\}\rightarrow \mathbb{R}##."

... ... so in defining a set of triples or 3-tuples we are dealing with a set of functions, one function for every 3-tuple ... is that correct?
One ##3##-tuple can be seen as one function. So yes, a set of ##3##-tuples can be seen as a set of functions this way.

(Mind you I guess it is as Munkres was defining a (one) J-tuple ... rather than a function that described a set f J-tuples ,,,)
Indeed, Munkres defined a ##J##-tuple simply as a function.
 

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