Simple set theory problem - definition of a J-Tuple

[Math Amateur]

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

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

 

[micromass]

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.

 

[Math Amateur]

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

 

[micromass]

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.

 

[blue_raver22]

A J-tuple is an ordered collection of elements from a set X, where the number of elements in the collection is determined by the set J. In other words, a J-tuple is a function that maps elements of J to elements of X.

In your example, J represents the indices or labels for the elements in the tuple, while X represents the set of possible elements. So, a J-tuple would be a mapping from the set J = {1,2,3} to the set X = {a,b,c,...z}, where each element in J corresponds to an element in X.

To create a concrete example, we can use different sets for J and X. Let's say J = {1,2,3,4} and X = {red, green, blue, yellow}. Then, a J-tuple would be a function that maps 1 to red, 2 to green, 3 to blue, and 4 to yellow. This would give us an ordered collection (or tuple) of four elements, (red, green, blue, yellow).

In this way, a J-tuple allows us to specify a specific number of elements from a set, and their order is determined by the set J. It is a useful concept in set theory and topology, as it helps us define and study ordered sets and their properties.