MHB Introduction to J-tuples in set theory

AI Thread Summary
The discussion centers on understanding J-tuples as defined by Munkres in topology, specifically the function x: J → X. The original poster struggles with the concept, providing an example where J = {1, 2, 3} and X is the alphabet, but finds it does not define a set of triples. Responses suggest clarifying the definition of a triple and propose exercises to illustrate the relationship between tuples and functions. The conversation emphasizes that a J-tuple is a function from J to X, where the codomain can vary based on the input, distinguishing it from standard tuples. The thread concludes with an exploration of the implications of this definition in broader contexts, such as dependent products in type theory.
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On page 113 Munkres (Topology: Second Edition) defines a J-tuple as follows:

https://www.physicsforums.com/attachments/2153

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

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

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

X = \{ a, b, c, ... \ ... \ z \} 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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Peter said:
On page 113 Munkres (Topology: Second Edition) defines a J-tuple as follows:

https://www.physicsforums.com/attachments/2153

I was somewhat perplexed when I tried to completely understand the function $$ x \ : \ J \to X $$.

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

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

$$ X = \{ a, b, c, ... \ ... \ z \} $$ then ...

obvioulsy 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
Hello Peter. I think I can help you but your question is not clear to me. Also, the LaTeX hasn't rendered properly. Can you please edit your post to render LaTeX better?

I am not sure what you mean by "... a map like 1-->a, 2--> d, 3--> h, does not work.."
Can you please elaborate?
 
caffeinemachine said:
Hello Peter. I think I can help you but your question is not clear to me. Also, the LaTeX hasn't rendered properly. Can you please edit your post to render LaTeX better?

I am not sure what you mean by "... a map like 1-->a, 2--> d, 3--> h, does not work.."
Can you please elaborate?

Thanks caffeinemachine

I have re-edited the post ... hope it now reads better.

Since x is a map from set J to set X i.e. \ x \ : \ J \to X) I just mapped the three elements of J to three particular elements of X

That is, I imagined, as an example a mapping by x such that

1 mapped to a

2 mapped to d

3 mapped to h

However, although this is a mapping from J to X under the function x it does not really define a set of triples.

Hope I am being clear ... hope you can help anyway ...

Peter
 
Peter said:
Thanks caffeinemachine

I have re-edited the post ... hope it now reads better.

Since x is a map from set J to set X i.e. \ x \ : \ J \to X) I just mapped the three elements of J to three particular elements of X

That is, I imagined, as an example a mapping by x such that

1 mapped to a

2 mapped to d

3 mapped to h

However, although this is a mapping from J to X under the function x it does not really define a set of triples.

Hope I am being clear ... hope you can help anyway ...

Peter
I think I understand your query.

First you should try defining a triple. In fact, you should also post the definition of a triple (in general an $n$-tuple) you are using.

And now try doing the following exercise.

Let $X$ be a set (not necessarily finite) and $T_3(X)$ be the set of all the triples of $X$. Let $F_3(X)$ be the set of all functions $\mathbf x:\{1,2,3\}\to X$.
Show that there is a bijection between $T_3(X)$ and $F_3(X)$.

The above exercise can easily be generalized for $n$-tuples.

After doing this, a natural question arises. So far we have only talked about $n$-tuples, where $n$ is finite. Can we make $\mathbb N$-tuples.. or $\mathbb R$-tuples? Or can we make $J$-tuples where $J$ is an arbitrary set?

You should be convinced that the definition of a $J$-tuple you have posted in the OP is a natural one.
 
Let's work with a really simple example.

Suppose we have a 2-element set, $X = \{a,b\}$.

Explicitly, the set of all pairs of elements from $X$ is:

$\{(a,a),(a,b),(b,a),(b,b)\}$.

Now let's look at the set of all functions:

$f:\{1,2\} \to X$. Explicitly, these are $\{f_1,f_2,f_3,f_4\}$, where:

$f_1(1) = a, f_1(2) = a$,
$f_2(1) = a, f_2(2) = b$,
$f_3(1) = b, f_3(2) = a$,
$f_4(1) = b, f_4(2) = b$.

Now it's clear that these sets have the same cardinality, so obviously we can make several bijections between them. But I have a SPECIAL bijection in mind:

Define $\phi:X^{\{1,2\}} \to X^2$ by:

$\phi(f) = (f(1),f(2))$.

We can write this as:

$\phi(f) = (x_1,x_2)$ where $x_i$ is BY DEFINITION, $f(i)$.

Doing this in THIS way, makes it clear that all we need is for $i$ to be in some set we can define our functions $f$ on, although we cannot realistically "list" these "coordinates" if the domain $I$ (which is $J$ in your text) is larger than countably infinite.

For example, we could "index" some sets by the real numbers, an example would be the open intervals $(a,\infty)$ where we have the functions:

$f:\Bbb R \to \mathcal{P}(\Bbb R)$ given by:

$f(a) = (a,\infty)$. We might "tag" these intervals as $I_a$.
 
Peter said:
However, although this is a mapping from J to X under the function x it does not really define a set of triples.
The definition says that an individual tuple is a function from $J$ to $X$. That is, such function does not define a set of tuples, it is a single tuple.

The unusual thing about tuples as functions is that their codomain depends on the argument: $x(\alpha)$ belongs to $A_{\alpha}$ and not simply $X=\bigcup_{\alpha\in J}A_{\alpha}$. In type theory, Cartesian product of different sets is called a dependent product, to distinguish it from $A^n$. Elements of the latter are regular functions from $\{1,\dots,n\}$ to $A$.
 
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