Introduction to J-tuples in set theory

[Math Amateur]

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

 

[caffeinemachine]

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 \(\displaystyle x \ : \ J \to X \).

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

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

\(\displaystyle X = \{ a, b, c, ... \ ... \ z \} \) 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?

 

[Math Amateur]

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

 

[caffeinemachine]

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.

 

[Deveno]

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 $I_a$.

 

[Evgeny.Makarov]

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 $A$.