# Simple set theory problem - definition of a J-Tuple

• Math Amateur
In summary: This definition can be applied to any set ##X##, not only ##\mathbb{R}##.In summary, Munkres defines a J-tuple as a function ##f:J \to X##, where J is the index set and X is the target set. This means that a J-tuple is essentially a set of functions, with each function representing a specific tuple. This definition can be applied to any set, not just the real numbers.
Math Amateur
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MHB
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

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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.

1 person
micromass said:
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

Math Amateur said:
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.

1 person

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.

## 1. What is a J-tuple in set theory?

A J-tuple in set theory is a sequence of J elements, where J is a positive integer. It is similar to an ordered pair, but instead of two elements, it can contain any number of elements depending on the value of J.

## 2. How is a J-tuple different from a set?

A J-tuple is different from a set in that it is an ordered sequence of elements, while a set is an unordered collection of distinct elements. Additionally, a J-tuple can contain multiple occurrences of the same element, while a set cannot.

## 3. What is the cardinality of a J-tuple?

The cardinality of a J-tuple is equal to the value of J, as it represents the number of elements in the tuple. For example, a 3-tuple would have a cardinality of 3.

## 4. How is a J-tuple represented mathematically?

A J-tuple can be represented mathematically as (a1, a2, ..., aj-1, aj) where a1 to aj are the elements of the J-tuple.

## 5. What are some real-life applications of J-tuples?

J-tuples have many applications in computer science and data analysis. They are commonly used in programming languages to represent data structures such as arrays and lists. They are also used in database management systems to store and retrieve information in a structured manner. J-tuples can also be used in mathematical modeling to represent ordered sequences of data.

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