I know this question has been asked like 1000 times, however all supplied answers were not really satisfying to me.
My question concerns the similarities and differences between these mathematical objects.
First, the Set. A set is defined to be the entity of distinct objects (not necessarily numbers). The arrangement of objects is not relevant. We use curly braces to denote sets; commata are used to seperate the objects within the set.
Second, the $n$-Tuple. A $n$-tuple is very similar to a set, however the objects need not to be the same and the ordering of objects within the $n$-tuple is important. $n$-Tuples are usually denoted with parentheses and the objects within are seperated with commata as in sets. Also, it is common to build the set of even numbers for instance like this: $\{2n\mid n\in \mathbb{N}\}$. However, I have never seen something like this with regard to n-tuples.
Third, the Vector. A vector is an element of a vector space. However, if I calculate the Cartesian product of, for instance, $\mathbb{R}×\mathbb{R}$ then the objects of $\mathbb{R}^2$ are (column-)vectors which are denoted as tuples. Furthermore, I often see box brackets to denote such vectors and the elements are written in one column (opposed to tuples or sets). Also, commata are not used to separate the objects (however, sometimes I see the elements of row vectors separated by commata). However, I have never seen such notation when for instance describing elements of $\mathbb{N}\times\mathbb{R}$.
Finally, matrices. Matrices are arrays of numbers and clearly linked to vectors as each column/row is a vector. However, I have never seen commata used in combination with matrices. Furthermore, the space of matrices is written as $A^{(m×n)}$. I know what the idea behind this notation is, however, as matrices are linked to vectors I have problems to really understand it.
Those concepts are obviously linked, however at certain points there arise crucial differences between them (which also come, I believe, from notational differences between authors and fields of mathematics). I hope my problem is comprehensible and someone can help me and shed light on my issues.
Thanks!
2 Answers
Preliminary Notions:
I would like to start by mentioning the fact that the terms set, tuple, vector, and matrix, are fairly high level abstractions that have come to be linked to somewhat generic notions across multiple sub-fields of mathematics, physics, and computer science. As a result, the laymen definitions of these objects are widely available, while formal definitions remain difficult to ascertain. This is especially true if you're aim is to have these formal definitions all reside within the same formal system. This brings us to our first problem: The formal definition of any given mathematical object really only holds water in the axiomatic or formal system within which it is defined. For example, Wikipedia says that:
"In mathematics, an n-tuple is a sequence (or ordered list) of n elements, where n is a non-negative integer."
However, in many systems, a sequence $a_n$ is precisely defined as a total function $a:\mathbb{N}\to\mathbb{R}$. This definition of sequence, combined with the definition of tuple in the quote above, implies that every tuple has a countably infinite number of entries. This, of course, is not a useful definition of tuple. The problem here is that we are mixing and matching the operational definitions of objects from different formal systems. I will now describe one possible way (in terms of sets) of formally relating all of the objects you mentioned, and try to answer all of your questions.
