Do tuples exist which aren't elements of a cartesian product of sets?

In summary, the conversation discusses the existence of tuples that are not elements of a cartesian product of sets. It is confirmed that such tuples do exist and can be defined through sets. However, there is also an alternative approach where tuples are defined as closer to the foundations of mathematics than sets. This idea is not widely accepted in the mathematics community, but some believe it could be beneficial in the age of computers and technology.
  • #1
bentley4
66
0
Do tuples exist which aren't elements of a cartesian product of sets?
Can you just write an ordered list of elements which does not necessarily have to be defined in sets? (or does every tuple need to be defined through sets in order for it to rigourously exist in mathematics?)
 
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  • #2
it's not clear what you're asking.

normally, an ordered tuple is a subset of SOME set AxBxC, but the sets involved might be unusual.
 
  • #3
The answer is yes.

Suppose you give me the triple (a, b, c). Then this is an element (in fact, the only element) of {a}x{b}x{c}.

This works for all (finite) tuples. (For infinite tuples, which are actually just sequences, we need to assume the axiom of choice to guarantee that there such a set exists.)
 
  • #4
bentley4 said:
Can you just write an ordered list of elements which does not necessarily have to be defined in sets? (or does every tuple need to be defined through sets in order for it to rigourously exist in mathematics?)
An natural question, particularly for a mathematically interested programmer: Wheras defining arrays (i.e. tuples) of any length is commonplace in all sensible programming languages, sets are absent from wide spread languages such as C.
Actually one can build the foundations of mathematics in a way that tuples are closer to the grounds than sets.
Most professors in mathematics probably feel that it would create more trouble than benefits if one would deviate from the narrow set based presentation of mathematics in courses.
Others feel that the strengthening role of computers, computing, and computation in science asks for a redesign of the taught foundations of mathematics.

You are on a good way; keep your eyes open an look behind the omnipresent orthodoxy.
 
  • #5


I can confirm that tuples do not necessarily have to be defined in sets in order to exist. A tuple is simply a finite ordered list of elements, and it can exist independently of any set. While tuples are often used in the context of set theory and cartesian products, they can also exist in other mathematical structures such as vector spaces and matrices. Therefore, it is possible for tuples to exist that are not elements of a cartesian product of sets. However, in order for a tuple to be rigorously defined in mathematics, it must have well-defined elements and a specific order. This can be achieved through sets, but it is not a requirement.
 

1. What are tuples?

Tuples are ordered collections of elements, typically enclosed in parentheses, separated by commas. They can contain any type of data, such as numbers, strings, or other data structures.

2. What is a cartesian product of sets?

A cartesian product is a mathematical operation that combines the elements of two or more sets to create a new set. The resulting set contains all possible combinations of elements from the original sets.

3. Can tuples exist outside of a cartesian product?

Yes, tuples can exist independently of a cartesian product. While tuples are often used in cartesian products as a way to represent ordered pairs or n-tuples, they can also exist as standalone data structures in programming languages.

4. How do I know if a tuple is an element of a cartesian product of sets?

To determine if a tuple is an element of a cartesian product, you can check if its elements are contained in the corresponding sets. For example, a tuple (1, 2) would be an element of the cartesian product of sets {1, 2} and {1, 2, 3}, but not of {1, 2, 3} and {2, 3, 4}.

5. Are there any restrictions on the elements of a tuple in a cartesian product?

No, there are no restrictions on the elements of a tuple in a cartesian product. As long as the elements are contained within the corresponding sets, they can be any type of data. However, some programming languages may have limitations on the data types that can be used in tuples.

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