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Simple set theory problem - definition of a J-Tuple

  1. Mar 21, 2014 #1
    On page 113 Munkres (Topology: Second Edition) defines a J-tuple as follows:

    attachment.php?attachmentid=67873&stc=1&d=1395452460.jpg

    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
     

    Attached Files:

  2. jcsd
  3. Mar 21, 2014 #2

    micromass

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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.
     
  4. Mar 21, 2014 #3
    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
     
  5. Mar 22, 2014 #4

    micromass

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    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.

    Indeed, Munkres defined a ##J##-tuple simply as a function.
     
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